Control of a dynamic brake to reduce turbine-generator shaft transient torques
by Matthew Kenneth Donnelly
A thesis submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in
Electrical Engineering
Montana State University
© Copyright by Matthew Kenneth Donnelly (1991)
Abstract:
The feasibility of applying a resistive dynamic brake to damp turbine shaft torsional oscillations is
studied. In cases where series-compensated transmission lines increase the likelihood of
subsynchronous resonance phenomena, the damping of one or more of the torsional modes of the shaft
may prove to be inadequate. This is particularly so when high-speed reclosing of transmission lines
near a generator is practiced. In this thesis, various control strategies and brake configurations are
studied and simulation results are shown. The simulations are performed using the ElectroMagnetic
Transients Program (EMTP) on models ranging from one machine, infinite bus (the IEEE Second
Benchmark) to multiple machines in a complex network. A derivative of Prony's method for signal
analysis is used to find a linearized transfer function of the system for controller design and
implementation. Methods for reducing the complexity of the results of a Prony analysis are presented.
It is seen both empirically and intuitively that a simple control law does an effective job of damping
turbine shaft torques throughout the range of subsynchronous frequencies. The empirical evidence is
gained through comparisons with a Generalized Predictive Control (GPC) algorithm implemented on a
linearized model of the test system. Discrete-level GPC is seen to have sensitivity problems in this
application. The thesis goes on to analyze and document these sensitivity problems. Recommendations
are made as to size, placement, and construction of the brake for adequate damping. Studies
incorporating multiple brakes are also conducted revealing that, with few exceptions, the interaction
between brakes operating on non-identical machines is negligible and that the interaction between
brakes operating on identical machines is substantial and desirable. Computer code for implementing a
brake in the EMTP is included as an appendix. 
Control of a Dynamic Brake to Reduce Turbine-Generator
Shaft Transient Torques
■>
by
Matthew Kenneth Donnelly
A thesis submitted in partial fulfillment 
of the requirements for the degree
of
Doctor of Philosophy 
' in
Electrical Engineering
MONTANA STATE UNIVERSITY 
Bozeman, Montana
November 1991
©  COPYRIGHT
by
Matthew Kenneth Donnelly 
1991
All Rights Reserved
0 ^ 7 ?
THIS'S'
ii
APPROVAL
of a thesis submitted by-
Matthew K. Donnelly
This thesis has been read by each member of the thesis 
committee and has been found to be satisfactory regarding 
content, English usage, format, citations, bibliographic 
style, and consistency, and is ready for submission to the 
College of Graduate Studies.
/3, /9 9 I
Date / 1 S Chairman/ Graduate Committee
Approved for the Major Department
Date V Head, Major department
Approved for the College of Graduate Studies
Date Graduate Dean
iii
STATEMENT OF PERMISSION TO USE
In presenting this thesis in partial fulfillment of the 
requirements for a doctoral'degree at Montana State University, 
I agree that the Library shall make it available to borrowers 
under the rules of the Library. I further agree that copying 
of this thesis is allowable only for scholarly purposes, 
consistent with "fair use" as prescribed in the U.S. Copyright 
Law. Requests for extensive copying or reproduction of this 
thesis should be referred to University Microfilms Interna­
tional, 300 North Zeeb Road, Ann Arbor, Michigan 48106, to 
whom I have granted "the exclusive right to reproduce and 
distribute copies of the dissertation in and from microfilm 
and the right to reproduce and distribute by abstract in any 
format."
IrI. I cIcIl
iv
ACKNOWLEDGMENTS
My sincere gratitude is extended to my thesis advisor. 
Dr. Roy Johnson. His faith in my abilities has motivated me 
in every phase of my research, and his personal friendship 
will be valued for many years to come. The help of Dr. Jim 
Smith and Dr. Don Pierre in the areas of system identification 
and control system design is also acknowledged. Further, I 
am indebted to Dr. John Lund and Dr. Ken Bowers for sparking 
in me an intense interest in the mathematical sciences which 
will stay with me forever.
In addition to the funding provided by the Electric Power 
Research Institute, I would like to acknowledge the financial 
contributions made by the Montana Electric Power Research 
Affiliates (MEPRA) and the Montana State University Engi­
neering Experiment Station (EES). The guidance provided by 
my industrial advisors is also appreciated. Mr. Ray Brush 
of Montana Power Company supplied me with much needed practical 
information. Dr. John Hauer of Bonneville Power Administration 
provided insight at our meetings and has been an inspiration 
in developing some system identification routines.
Lastly, I would like to thank my fiance', Toni Crosby, 
for enduring the long days and late nights associated with 
graduate study. Her interest in the work and her support and 
understanding have made this research enjoyable.
TABLE OF CONTENTS
LIST OF TABLES .............................. ......... vii
LIST OF FIGURES .....   viii
NOMENCLATURE ............'............................. xi
ABSTRACT ...............................................xii
1. INTRODUCTION ............. '........................ I
Histoiy Of The Problem ........................2
Description Of The Problem ................... 3
Pertinent Literature ..........................  7
2. MODELS AND MODELING .............................. 10
The MPC Colstrip Model .... ................... 12
The Dynamic Brake ......................... . . . . 15
Implementation Of The Brake In EMTP ..... . 16
An Alternative Integration Technique for
Time-Domain Simulation .................  20
3. SYSTEM ANALYSIS TECHNIQUES ..... .................  24
Frequency Scan .................................24
Electromagnetic Transients Program (EMTP) .... 28
Eigenvalue Analysis ............... . . . *...... 29
System Identification And Modeling ........... 29
Output Analysis of Prony
Identification Routine .................  32
4. CONTROLLER DESIGN ....... :........................39
Straight Feedback Control ............... . ....41
Filtering The Feedback Signal ................  46
Optimal Control ...............................  48
Sensitivity Problems with
Discrete-Level GPC ......................51
Sizing The Brake . ....... ......................66
V
TABLE OF CONTENTS —  Continued
5. SIMULATION RESULTS ............. . ................. 70
Effects Of Filtering ..........................  71
Results With A 200 MW Brake .................. 75
Effects Of Brake Size .........................77
Damping Electromechanical Modes ............. 80
Thyristor Types .......................... . 82
Effects Of Braking On Nearby Machines .■...... 84
6. EFFECTS OF FAULT TYPE, POWER FACTOR, AND BRAKE
ORGANIZATION ......................................  87
Multiple Brakes At A Single Location .......... 94
A Single Brake For Multiple Machines .......... 97
7. CONCLUSIONS AND FUTURE WORK ....................... 101
REFERENCES CITED ......................................  109
APPENDICES ...................................... 116
A-IEEE SECOND SUBSYNCHRONOUS BENCHMARK DATA . . 117 
B-SYSTEM DATA FOR MULTI-MACHINE'SIMULATIONS .. 122
C-EMTP INTERFACE AND CONTROLLER CODE .......... 142
D-ADAMS PREDICTOR-CORRECTOR TO IMPLEMENT A
PASC SYSTEM .............    147
vi
vii
LIST OF TABLES
Table < Page
I. Prony Analysis Results for Example System ....  35
viii
LIST OF FIGURES
Figure Page
I. The IEEE Second Subsynchronous Benchmark .....  11
2 . MPC Colstrip Model ...... ........................14
3 . Wye Connected Dynamic Brake .................... 15
4. Dynamic Brake With Type 60 Sources .......... 17
5. EMTP Calls To Control Subroutine .............. 20
6. Frequency Scan From Colstrip #1 .............. .  2 6
7. Frequency Scan From Colstrip #3 ............... 27
8. Eigenvalues of the MPC System From Colstrip #3 31
9 . Errors Between Actual and Estimated Response
for the System in (3.4) ........................ 36
10. Root Locus of the MPC Colstrip Test System at 
Colstrip #3 Using Generator Aco as a Feedback 
Signal .......................................... 42
II. Root Locus of the MPC Colstrip Test System at
Colstrip #3 Using Generator Acceleration as a 
Feedback Signal .... ■......................... ..44
12 . Root LocuS of the MPC Colstrip Test System at
Colstrip #3 Using High Pressure Turbine Aco as a 
Feedback Signal ................................  45
13 . Root Locus of the MPC Colstrip Test System at
Colstrip #3 Using the Filtered Generator Aco as 
a Feedback Signal ..............................  47
14. Uncontrolled Response of System Due to Pulse
Disturbances .................................... 66
15. Damping of System From Feedback Gain and GPC .. 66
Figure Page
16. Second Benchmark Generator Speed Deviation .... 67
17. Second Benchmark Gen-LP Torque ................. 67
18. Uncontrolled Generator Aco for Three Phase Fault
with Reclose .....................................73
19. Filtered and Unfiltered Controlled Response for
Case in Figure 18 ............................. 73
20. Generator Speed Deviation and Brake Current for
Unfiltered Controller ..........................  74
21. Generator Speed Deviation and Brake Current for
High-Pass Filtered Controller .................  74
22. LP1-LP2 Torque for Disturbance in Figure 18
Showing Effect of Filtering ...................  75
23. HP-IP Torque for Disturbance in Figure 18 ....  76
24. LP2-Gen Torque for Disturbance in Figure 18 ... 77
25. LP1-LP2 Torque for Disturbance in Figure 18
Using 800 MW Brake ............................. 7 9
26. HP-IP Torque for Disturbance in Figure 18 Using
800 MW Brake .............................. ; . . . . 80
27. Generator Aco Signals Showing the Electromechan­
ical Mode Damping .............................. 82
28. Generator Ago Signals Showing the Effect of
Thyristor Type ..................................83
29. Colstrip #1 Uncontrolled Generator Speed Devi­
ation ......................................... . . 85
30. Colstrip #1 Generator Speed Deviation for 200
MW and 800 MW Brake ............................ 85
31. FFT of Generator Ago ............................ 89
32 . FFT of HP-IP Torque ............................. 89
ix
LIST OF FIGURES —  Continued
XFigure Page
33 . FFT of IP-LPl Torque ............................ 89
34. FFT of LP1-LP2 Torque .......................... 89
35. FFT of LP2-Gen Torque ........................... 90
36. Generator Aco Signals Showing Damping of SLG
Fault ........................................... 91
LIST OF FIGURES —  Continued
37. LP1-LP2 Torque Showing Damping of SLG Fault ... 92
38. LP2-Gen Torque Showing Damping of SLG Fault ... 93
39. Colstrip #3 Ago Signals Showing Damping for
Multiple Brakes at a Single Location .........■. 95
40. Colstrip #1 Aco Signals Showing Damping for
Multiple Brakes at a Single Location .......... 96
41. Colstrip #3 LP1-LP2 Torque Showing Damping for
Multiple Brakes at a Single Location .......... 96
42. Colstrip #1 LP-Gen Torque Showing Damping for
Multiple Brakes at a Single Location . . . ...... 97
43 . Colstrip #3 Aeo Signals Showing Damping for a
Single Brake on Multiple Machines .............  98
44. Colstrip #1 Aco Signals Showing Damping for
Multiple Brakes at a Single Location .......... 99
45. EMTP Data File for IEEE Benchmark .......   118
46. EMTP Data File for Multi-Machine System ........123
47. Controller Subroutine INJECT .................... 143
48. Modifications to the EMTP Subroutine CSUP .....  144
49. Adams Predictor-Corrector for Experimental PASC
System ........................... 148
50. File to Execute the Integration Routine of
Figure 49   165
xi
NOMENCLATURE
Lf The LaPlace. transform of f
^ 1S The ■inverse LaPlace transform of g
Il Il Any induced matrix norm
K(A) The condition number of A
B+ The Moore-Penrose Inverse of B
Q h The conjugate-transpose of Q
xii
ABSTRACT
The feasibility of applying a resistive dynamic brake to 
damp turbine shaft torsional oscillations is studied. In 
cases where series-compensated transmission lines increase 
the likelihood of subsynchronous resonance phenomena, the 
damping of one or more of the torsional modes of the shaft 
may prove to be inadequate. This is particularly so when 
high-speed reclosing of transmission lines near a generator 
is practiced. In this thesis, various control strategies and 
brake configurations are studied and simulation results are 
shown. The simulations are performed using the ElectroMagnetic 
Transients Program (EMTP) on models ranging from one machine, 
infinite bus (the IEEE Second Benchmark) to multiple machines 
in a complex network. A derivative of Prony's method for 
signal analysis is used to find a linearized transfer function 
of the system for controller design and implementation. 
Methods for reducing the complexity of the results of a Prony 
analysis are presented. It is seen both empirically and 
intuitively that a simple control law does an effective job 
of damping turbine shaft torques throughout the range of 
subsynchronous frequencies. The empirical evidence is gained 
through comparisons with a Generalized Predictive Control 
(GPC) algorithm implemented on a linearized model of the test 
system. Discrete-level GPC is seen to have sensitivity 
problems in this application. The thesis goes on to analyze 
and document these sensitivity problems. Recommendations are 
made as to size, placement, and construction of the brake for 
adequate damping. Studies incorporating multiple brakes are 
also conducted revealing that, with few exceptions, the 
interaction between brakes operating on non-identical machines 
is negligible and that the interaction between brakes operating 
on identical machines is substantial and desirable. Computer 
code for implementing a brake in the EMTP is included as an 
appendix.
ICHAPTER I 
INTRODUCTION
The primary focus of this thesis is the damping of transient 
torques in turbine-generator shafts following electrical 
network system disturbances. The damping is to be improved 
by means of a dynamic resistive brake located near to the 
generator. Transient torques are oscillatory in nature. It 
is desirable to damp these oscillations as quickly as possible, 
particularly in the case where high speed reclosing is employed 
on transmission lines emanating from the generator.
This dissertation contains seven chapters. Chapter I is 
an introduction to the problem along with a brief history and 
literature review. Chapter 2 describes the models used in 
the research and the associated modeling techniques. Because 
of the complexity of the electrical network and the cost of 
lost generation, accurate and realistic modeling is a crucial 
task. Therefore, a significant amount of work was spent in 
developing appropriate models and simulation routines. This 
effort should prove to be a great benefit to other researchers 
in this area. Chapter 3 discusses some of the techniques 
that are useful in the analysis of subsynchronous resonance
2problems. Specifically, the analysis techniques used in 
determining appropriate feedback signals for the dynamic brake 
and for sizing the brake are explained. Chapters 4 and 5 
describe the control techniques, controller analysis and 
simulation results. In Chapter 6, some special cases are 
examined to determine the effectiveness of the control 
algorithm over a wide range of conditions. Finally, Chapter 
I offers some conclusions relating to the project and provides 
some direction for future work. Following Chapter 7, a list 
of references is provided.
In addition to the text, four appendices are included 
showing some representative computer code used in this 
research. Appendices A and B contain data files for the "IEEE 
Second Subsynchronous Benchmark" model and the Montana Power 
Company Colstrip model. Appendix C shows the interface and 
controller subroutines. Appendix D contains computer code 
implementing a numerical integration technique along with a 
sample data file described in Chapter 2.
History of the Problem
The notion of series compensation to increase the carrying 
capacity in electric power transmission lines beyond what is 
considered the uncompensated stability limit was developed 
almost simultaneously with the widespread adoption of 
alternating current as the transmission standard in the US.
3Dr. Charles Concordia recognized as early as 1937 that series 
compensation would set up a resonance in the electrical network 
and that this was a potential problem [1]. Dr. Concordia and 
other early researchers saw this resonance as a problem only 
with respect to the electrical network. It was not until 33 
years later when a turbine-generator was actually damaged in 
operation that researchers began to look at the interaction 
between the electrical network and the turbine-generator shaft 
mechanical system.
In 1970, the first of two shaft failures occurred at the 
Mohave plant in southern Nevada. By 1972, a consortium of 
engineers had determined that the cause of the 197 0 shaft 
failure and the second failure in 1971 were due to interactions 
between the electrical network and the mechanical system at 
certain frequencies where both systems showed a tendency to 
resonate [2], [3]. This phenomenon has been the topic of 
much research since that time.
Description of the Problem
Power system network disturbances create several distinct 
transients in nearby generation units. One such transient 
occurs in steam-driven generators as the separate masses along 
the turbine-generator (TG) shaft oscillate against each other. 
The magnitude of the inertias of these masses along with the
4spring constant of the connecting shaft determine the fre­
quencies of oscillation according to Newtons Law for torsional 
mechanics,
Torque =Inertiaxangularacceleration.
Current technology in the field of turbine design is such 
that these natural frequencies of oscillation fall into the 
range of 10 to 50 Hertz. Power system transmission networks 
are relatively poorly damped at frequencies below synchronous 
speed. In some cases the combination of the network and 
turbine shaft resonances at a frequency can be negatively 
damped. It is this interaction between the poorly damped 
network and the turbine shaft natural frequencies that has 
been described as Subsynchronous Resonance or SSR.
The phenomena of subsynchronous resonance can be divided 
into two main areas for the purpose of analysis. These are 
generally recognized in the literature as being self­
excitation phenomena and transient torque phenomena. The 
area of self-excitation effects is often further divided into 
induction motor effects and torsional interactions. In the 
cases that have been reported, both conditions occur simul­
taneously [4] . From a control perspective self-excitation 
phenomena is not a separate issue from transient torque 
phenomena but rather the issue is how these electrical and 
mechanical interactions affect the damping of turbine shaft 
oscillations. A shaft system with very low or no damping
5will exhibit steady-state instability. A system with larger 
damping factors for each natural mode will not exhibit steady 
state oscillations, but may experience damaging shaft torques 
for a given sequence of network events.
The practice of transmission line high-speed reclosing 
may fatigue some turbine-generator shaft systems [5], [6]. 
Although the shaft is sufficiently damped so as to prevent 
steady state oscillation, the damping factors are small enough 
that damaging shaft torques can be created by a high speed 
reclosing operation. A system with natural modes of oscil­
lation will always respond more actively to inputs that contain 
frequencies near to the natural modes. High-speed reclosing 
involves faster operation of breakers which necessarily means 
creating shocks to a shaft system that contain a larger amount 
of energy at frequencies nearer to the natural shaft modes.
Generator shaft torsional damping results from both 
mechanical and electrical factors. The mechanical factors 
include windage, bearing friction, and natural damping of the 
shaft material. The windage on turbine blades is cited as 
the major factor which causes shaft systems to exhibit more 
damping at higher machine loadings [7]. Electrical factors 
contributing to damping are predominated by the electrical 
resonance points of the network. Maintaining a high electrical 
impedance at all frequencies within the SSR range will 
eliminate problems of steady state shaft oscillations but may
6not be sufficient to produce enough damping to safely allow 
for high speed reclosing. This may be an important consid­
eration for a control strategy which is intended to produce 
enough damping to allow for high speed reclosure. Such a 
strategy will likely have to provide much more damping than 
would be obtained passively by changing the apparent network 
impedance over the range of subsynchronous frequencies.
It is also important to realize that during a typical 
short circuit disturbance with fault clearing and reclosing, 
the generator shaft system is exposed to three suddenly 
different electrical networks. These correspond to the 
apparent impedances seen during the short, those seen during 
clearing and those seen upon reclosure. These three networks 
are very likely to have different resonant characteristics 
and result in different damping factors for each of the shaft 
natural modes. With this in mind a controller that will 
produce good damping must be able to operate on the shaft 
system during as much of these three time periods as possible.
This study examines the use of a resistive brake at the 
terminals of a large synchronous generator which is activated 
in such a way and at such a time so as to provide additional 
damping to the turbine-generator shaft during critical 
transients. Only steam-powered generators (thermal units)
7are considered since hydro-electric generators do not exhibit 
mechanical oscillations in the frequency range of concern for 
this work.
Pertinent Literature
There is an abundance of literature dealing with the topic 
of subsynchronous resonance. The vast majority of these 
papers treat SSR as a sustained phenomenon and do not discuss 
the transient torque problem associated with turbine-generator 
shaft fatigue. The reader is referred to the IEEE working 
group bibliographies on SSR for an exhaustive list of ref­
erences in this area [8] - [11] . The IEEE working group has 
also published a reference of terms, definitions, and symbols 
relating to SSR phenomena [12] . In [13], Bowler gives a 
concise summary of the various aspects of SSR that will provide 
the reader with a familiarity of the topic without thorough 
research. In his text on power system dynamics [14] , Yu also 
gives an excellent summary. Hamouda, Iravani, and Hackam 
give insight into the specific problem of torsional oscil­
lations and provide an integrated study model in [15.] . Agrawal, 
et al., [16], explain some of the evaluation tools available 
to researchers in this field.
Several papers have investigated the use of braking 
technologies to improve power system stability in the 
"transient stability" sense. Traditionally, transient sta-
fM l i t y  has referred to the ability of synchronous generators 
to return to the system synchronous frequency following a 
major disturbance. In [17], [18], and [19], some optimal 
control strategies are developed to damp low frequency swings 
using braking resistors. These strategies are implemented 
in computer simulations to evaluate their effectiveness. 
Yoshida's work, [20], is more functional in that he discusses 
some implementation issues with respect to resistive devices 
in the network. A resistive brake acting alone may not provide 
the optimal amount of damping in transient stability studies. 
The coordination of system components such as exciters with 
a braking resistor is discussed in [21]. Of equal interest 
to the theoretical studies, the testing of braking resistors 
on the actual system is always enlightening. In [22] and 
[23] , the results of testing at two sites which have installed 
large braking resistors are discussed.
Additionally, several methods have been proposed to 
address the problem of damping transient torques in TG shafts. 
Hingorani, et al. discuss the possibility of using a series 
device to damp transient torques [24], [25]. The referenced 
device appears to be better suited for damping the sustained 
oscillations commonly associated with severe SSR. In [26], 
the generator field is adjusted to attempt to compensate for 
torsional oscillations with some success. The use of a
8
resistive brake to damp higher frequency oscillations such
9as can be seen in the mechanical mass-spring system of a 
turbine shaft was first researched at Purdue University [27] , 
[28] . No other recent literature has been found dealing 
specifically with this topic. Generally, the brake size may 
be smaller than would be seen in a brake designed to improve 
transient stability, but expensive thyristors must be used 
to switch the brake. The low frequencies involved in tra­
ditional transient stability studies allow the use of circuit 
breakers to switch these resistors.
10
CHAPTER 2
MODELS AND MODELING
The development, testing and verification of models and 
modeling techniques make up a large percentage of the work 
performed in this research.' This chapter provides the doc­
umentation for the two test systems that were developed for 
use in the study of transient torque. The information provided 
herein should be useful as a starting point in any future 
studies involving turbine-generator shaft oscillations and 
the transient torque phenomena.
A one-line diagram of the first test system examined is 
shown in Figure I. The basis for this model is the IEEE 
Second Subsynchronous Benchmark case as reported in [29] . 
This system provides a benchmark by which all studies in this 
area can be compared. The machine and line parameters found 
in the paper are implemented in an Electromagnetic Transients 
Program (EMTP) [32], [34], [37] . The damping factors for the 
generator shaft system in this study have been increased over 
what is specified for a self-excitation case. The actual 
damping values used are as specified in [29] under the section 
titled. "Torque Amplification Studies." The IEEE First
11
Subsynchronous Benchmark [30] was not deemed to be suitable 
for these studies because the authors made no provision for 
simulating transient SSR (torque) problems.
The Second Subsynchronous Benchmark model is useful in 
testing various control strategies on a single machine system 
and in comparing the effectiveness of one control technique 
over another. The system is fairly simple and as an IEEE 
benchmark it can lend credibility to simulation results. The 
basic EMTP data file used to implement this benchmark is shown 
in Appendix A.
BUS 2 BUS 1
4-MASS 
600 MVA H
— [ I— BUSO
BRAKE
Figure I . The IEEE Second Subsynchronous Benchmark.
A second test system developed for the braking studies 
is the Montana Power Company (MPC) Colstrip Model and asso­
ciated 500 kV transmission lines. This system has several
12
advantages over the benchmark system which make it more useful 
as the primary test bed for the dynamic brake stability studies 
presented here. The basis for the model was provided by Ray 
Brush of MPC in late 1989. Though significantly modified 
from the original MPC data for the purposes of this study, 
the current model has retained enough major features of the 
original to make it a realistic facsimile.
The MPC Colstrin Model
Three major modifications to the original model stand out 
as being significant. They are: I) the inclusion of the J . 
Marti line models [31] on the. 500 kV lines replacing n  
sections, 2) the elimination of the zinc oxide varistors used 
as series capacitor protection, and 3) creation of Thevenin 
equivalents at the 230 kV buses and at Dworshack and Bell 
substations. Numerous other less dramatic changes were made 
in converting the files to EMTP version 2.0 and tuning the 
simulations.' The current model is well suited to studies on 
transient torque and dynamic braking. A one-line diagram of 
the system is shown in Figure 2. The Colstrip generators 
appear on the right as synchronous machines and are labeled 
SM.
The model, implemented in the Electric Power Research 
Institute's (EPRI's) version of EMTP, consists of two sets 
of two identical turbine-generators for a total of four
13
machines at the generating facility. Two machines, Colstrip 
#1 and Colstrip #2, feed a 230 kV bus located at Colstrip and 
are each rated at 377 MVA. An autotransformer then feeds the 
500 kV bus at Colstrip from the 230 kV bus. These are three 
mass machines. The masses consist of two turbines and a 
generator. Colstrip #3 and Colstrip #4 are two larger machines, 
each rated at 818 MVA, that feed the 500 kV bus at Colstrip 
directly. These are six mass machines consisting of four 
turbines, a generator, and an exciter.
The machines can be configured in several different ways 
depending on the type of study desired. One common config­
uration in the preliminary studies was to combine the two 
large units into one 1636 MVA conglomerate machine and 
implement the brake at the terminals of this machine. In 
previous studies on the topic, the case with multiple machines 
-- particularly closely coupled machines -- has not been 
considered.
The bulk of the energy from Colstrip is carried on twin 
500 kV circuits, both of which are series compensated. The 
series compensation can provide a turbine-generator mass­
spring system with negative damping under certain circum­
stances as identified in numerous papers.on SSR. In almost 
all cases, the presence of series compensation on a 
transmission line has the effect of reducing the damping of 
at least one mode of a turbine-generators natural frequencies
HSSOO
TFSOO
DWSOO
BESOO
CS222CS230
BV230BVSOOyCASOO
'Ti '*CA23°
' BVTHV
CB341BC7S1
BCSW2BC7S2
H
Figure 2. MPC Colstrip Model
15
of oscillation. This reduction in damping can in turn create 
problems with transient torques in turbine-generator shafts 
following system disturbances. Again, the Purdue studies do 
not incorporate series compensation into their systems for 
studying transient torques.
The basic EMTP data file for the MPC Colstrip model is 
included in Appendix B .
The resistive brake is connected to the system in wye at 
the generator terminals. The wye connected brake is shown 
in Figure 3 .
The Dynamic Brake
A
B
C
GEN
Figure 3. Wye Connected Dynamic Brake.
Historically, resistive brakes have been placed at higher 
voltages than is seen at the terminals of a synchronous 
machine. The Peace River 600 MW brake is at 138 kV [22] and
16
the BPA 1400 MW brake is at 230 kV [23]. Since the primary 
function of these devices has been for improving transient 
stability, they are large and employ circuit breakers as 
switching•devices. The brake presented here utilizes thy­
ristors as switching devices, and so a lower operating voltage 
may be considered advantageous. Circuit breakers may still 
be employed in a normally closed position to protect the 
system against thyristor failure.
The size of the brake is determined by the value of 
resistance. In this study, brakes ranging from 41 MW to 850 
MW were used. It is difficult to put a percent rating on 
these values when dealing with multiple machines, however 850 
MW is approximately 50% of the conglomerate Colstrip machine 
rated MVA.
Implementation of the Brake in EMTP
The brake is implemented in the EMTP as shown in Figure 
4. The EMTP Type 60 sources connected in each phase to ground 
are TAGS controlled sources. TAGS is a sub-program of the 
EMTP that implements controllers on the simulated systems 
[32]. TAGS is an acronym for Transient Analysis of Control 
Systems. In this case, these sources are always proportional 
to the voltage at their respective generator terminals. The 
relationship between the generator bus voltage and the Type
17
60 source is
where
^TypeSO A = ^Bus a(1 ~ Vz7) 
^TypedO B ~  ^B u s fl(l ~ V^) 
^TypedO C = ^ Sus c(^  - V^)
0<M <1.
(2.1)
Type 60 Type 60 Type 60
Figure 4. Dynamic Brake With Type 60 Sources.
For "bang-bang" control of the type employed in [27], let 
M = I in (2.1) . Then the status of the thyristors only determine 
whether the brake is off or fully on. Bang-bang control 
refers to a control law which provides maximum control action 
for a period of time and then switches to minimum control 
action without taking on any intermediate values. Now for 
a value of m =0.5,
18
p  _(V s u s i  VfypedOi)
"  R
R
_ VBusi
~ 2R *
Should the thyristors now gate the brake, it would be supplying 
only one half of the potential, braking power. The Type 60 
source therefore gives the flexibility to design controllers 
of the conventional type as well as looking into bang-bang 
control.
Although nonlinear systems are in general difficult to 
analyze, they often can be accurately approximated by linear 
systems. A control strategy with the objective of damping 
oscillations in a minimum amount of time is referred to as 
time optimal control. Time optimal control of systems with 
limited control actuator energy are usually bang-bang con­
trollers [33]. For this reason, the most effective control 
strategy for damping shaft transient torques for fast reclosing 
will likely involve bang-bang control of the brake. The Type 
60 sources described above are used as a starting point for 
brake modeling and research into the possible uses of the 
brake in improving shaft damping and transient stability. 
In the bang-bang implementation, the Type 60 sources are 
unnecessary but are included for consistency.
19
Long term damping in the transient stability case may not 
be most readily obtained from bang-bang control. Thyristor 
firing angle control could be used to implement such a long 
term braking action in which case the Type 60 source concept 
may be an adequate model. If the controller called for u = 0.5 
in (2.1), the firing of the appropriate thyristor would be 
delayed long enough to admit only "half power" through its 
resistor in any given half cycle of the 60 Hz bus voltage.
The thyristor models and the Type 60 sources are each 
controlled in this study by TAGS supplemental variables. The 
determination of the Type 60 source amplitude and the firing 
angle of the thyristors are computed using various control 
strategies which will be discussed later. These calculations 
are conducted in a FORTRAN subroutine which is linked with 
the EMTP. The EMTP support subroutine CSUP is used to place 
the software interface "hooks" to the user written controller. 
Subroutine CSUP is called at each integration time step to 
calculate the values of some of the TAGS supplemental vari­
ables. Node voltages are brought in to the user written 
subroutine via three supplemental variables, one per phase, 
defined in the EMTP data file. Synchronous machine variables 
are brought in via the common block SMTACS with the array 
ETACS. With this information, the controller subroutine 
delivers the Type 60 source amplitudes and the status of the 
thyristor gate firing signals through other supplemental
20
variables defined in the EMTP data file. The array containing 
the supplemental variables at this point in the time step 
integration is called XTCS. A block diagram containing the 
pertinent calls at each time step is shown in Figure 5.
S U B T S "  ca lled  a t  
ea c h  tim e  s te p
BRAKE
,CONTROLLERUser Written
SUBTS1 SUBTS4SUBTS3
CSUP
SUBTS2
TACS3
OVER16
MAINIO
MAINOO
Figure 5. EMTP Calls To Control Subroutine.
A description and listing of the modifications made to 
the EMTP source code is included in Appendix C . The new 
external subroutine called from the EMTP contains the control 
algorithm and is discussed in Chapter 4.
An Alternative Integration Technique for 
Time-Domain Simulation
The EMTP uses a trapezoidal integration rule to solve the 
network equations representing the modeled system. This 
choice has historically been motivated by the robustness of 
the trapezoidal rule in the face of complexities often found 
in power system models such as nonlinearities, switching
21
actions which create time-domain discontinuities, and wave 
phenomena [34] . The trapezoidal rule can be placed in a 
broader category of numerical schemes known as Adams methods. 
The particular rule used in the EMTP is known as a second-order 
Adams-Moulton scheme. Some steps that are common in the 
development of all Adams methods are shown below [35].
Writing the differential equation
dx
(2.2)
in the form
dx -f{t,x)dt (2.3)
one integrates (2.3) between tk and tk+1
h+i f*+iJ dx=xk+l-xk= J f(t,x)dt . (2.4)
h h
The function f(t,x) is now approximated as a polynomial so 
that the integral on the right-hand side of (2.4) may be 
evaluated. If Newton-Gregory backward interpolating poly­
nomials fitting at tk,tk_k,...,tk_n are used to approximate f{t,x) 
and the integration in (2.4) is carried out, the result is 
a jfc + r  order Adams-Bashforth predictor scheme. Now if f(t,x) 
is approximated using a (k + l)st degree polynomial fit at 
4+1,4, -.,4-n+i/ a different scheme called an Adams -Moulton 
corrector is obtained.
22
The fourth-order Adams-Bashforth predictor is
(2.5)
and the fourth-order Adams-Moulton corrector is
xk - -1 + ^4 (9A  + 19A  -1 ~ 5/t - 2 + A  - 3) • (2.6)
In (2.5) and (2.6), h denotes the step size.
In standard practice, (2.5) would be applied to the set 
of system equations constituting the system model. Once xk 
is obtained from the predictor, it is used to estimate fk and 
(2.6) is applied to the system equations. The Xk obtained 
from the corrector is a better estimate of the actual value 
at tk than is xk.
The higher order predictor-corrector schemes provide more 
accurate estimates of the true system, but they suffer one 
major drawback. For the fourth-order scheme shown in (2.5) 
and (2.6) , four past values of / are required to initiate the 
simulation. Since it is already a difficult task to formulate 
a three-phase load-flow to find initial conditions for the 
trapezoidal rule, the higher order predictor-corrector schemes 
are not implemented in the EMTP.
There are cases ideally suited for predictor-corrector 
integration schemes. In some simulations, the user is 
interested in transients occurring during the start-up of a 
system as well as , the steady-state. In many cases, the
23
transients are fast and the steady-state can be reached in 
a minimum of computation time. In these situations, one can 
set all of the initial conditions to zero and start the 
simulation from rest. Such is the case with the " PASC" 
converter described in [36].
Appendix D contains the source code for a model of the 
PASC system overlaid on ' a generic implementation of a 
fourth-order Adams predictor-corrector. The PASC model proved 
to be difficult to simulate using some implementations of 
standard integration routines. The Adams routines had the 
advantage of being a-stable and simple. These are the features 
noted by the authors of the EMTP in defending their choice 
of integration routines also [37]. Should a greater amount 
of accuracy be required in the future, it may be worthwhile 
to re-write the EMTP to incorporate a higher order Adams 
scheme. Such a re-write is beyond the scope of the present 
work.
24
CHAPTER 3
SYSTEM ANALYSIS TECHNIQUES
There are several system analysis techniques which are 
suggested in the open literature for studies dealing with SSR 
phenomena. Three of these techniques, frequency scan, EMTP, 
and eigenvalue analysis, have been widely used in previous 
research on the topic. Agrawal et al. have done an excellent 
job of summarizing these in a paper submitted at the 1989 
IEEE Power Engineering Society Summer Meeting [16]. In 
addition to these conventional analysis techniques, system 
identification methods resulting in identified models have 
proven to be useful tools in the design of controllers for 
the brake. The identified models are obtained using an 
off-line system identification routine. The three conven­
tional methods and the identified model are discussed in more 
detail in this section.
Frequency Scan
Frequency scanning is a technique which can be used to 
identify frequencies in the electrical network that pose 
potential SSR problems. A frequency scan shows the equivalent
25
network resistance and reactance "looking out" from a point 
in the network as a function of frequency. For SSR analysis, 
this point in the network is invariably at the terminals of 
the generator under study. One can obtain a plot of resistance 
and reactance versus frequency for a particular generator and 
determine whether the electrical network, as seen from that 
generator, will exhibit any resonances within a certain 
frequency range -- typically 10 to 50 Hertz. A resonance 
occurs where the reactance at a frequency is identically zero. 
The effects of this network resonance on the stability of a 
turbine-generator shaft system are amplified if the network 
resistance at the resonant frequency is small or perhaps even 
negative. Agrawal and Farmer comment on the use of frequency 
scanning in SSR analysis in [38].
Commonly, the mechanical mass-spring system consisting 
of the turbine shaft is analyzed separately for the mechanical 
resonances. These frequencies are then reflected to the 
electrical network using frequency-domain convolution [see 
39, p .451]. Convolution is appropriate because the action 
of the generator mass oscillating at some frequency while 
rotating at synchronous speed modulates the magnetic flux 
acting on the machine stator. This process amounts to the 
equivalent of amplitude modulation in communication theory. 
The net effect of this convolution is the appearance of two 
"sideband" frequencies in the electrical network. One sideband
26
can be seen at ^ nchronom- ^ chanical and one can be seen at 
f synchronous+/mechanical* The latter frequency is of no interest in SSR 
studies due to the high network damping at supersynchronous 
frequencies [40]. The resulting subsynchronous frequency is 
normally called the "compliment" frequency.
The mechanical system compliment frequencies are drawn 
on the frequency scan plots as vertical lines for a visual 
aid in identifying potential problems. Figures 6.and 7 show 
frequency scans from the terminals of Colstrip #1 and Colstrip 
#3 respectively. These plots were obtained using the FREQUENCY 
SCAN option of the EMTP. As would be expected for machines 
which are electrically close together, the two plots are 
similar. Figure 6, however, contains more inductive reactance 
than does Figure 7. This is due to the presence of the 230/500 
kV auto-transformer between the two machines.
Frequency  scan  from  Colstrip  #1
0.04
Frequency
Figure 6. Frequency Scan From Colstrip #1.
27
Frequency  scan  from  Colstrip  #3
0.15
0.05
-0.05
Frequency
Figure 7. Frequency Scan From Colstrip #3.
Sharp fluctuations in the reactance curve indicate the 
presence of a network resonance at the corresponding frequency. 
This is analogous to the "Q" factor of an oscillator in circuit 
theory. In Figure 7, the points at which the reactance is 
zero are resonant points. Although the network as seen from 
Colstrip #1 (Figure 6) has no genuine resonant points, there 
may still be considerable energy present at certain fre­
quencies. This is important to remember for transient torque 
studies. Lightly damped oscillations of this nature in the 
network at one of the turbine shaft compliment frequencies 
would tend to prolong oscillations in the shaft itself.
Colstrip #1 has three masses and therefore three resonant 
frequencies or modes. One mode is always a result of all of 
the masses on a turbine-generator shaft oscillating together.
28
This mode creates no torque on the shaft and so is of no 
interest in these studies. The compliment of the two remaining 
modes are shown in Figure 6. The mode at approximately 26 
Hertz is seen to fall close to a network resonance. In Figure 
7, the modes at approximately 15 and 24 Hertz fall close to 
network resonant points.
Frequency scanning can be seen to be a useful tool in the 
preliminary analysis of SSR studies. By this analysis the 
Colstrip machines would seem to be fairly safe from SSR 
problems of the sustained type, but susceptible to high 
transient torques at one or more modes. It can also be seen 
that the auto-transformer plays an important attenuation role 
in the suppression of sustained SSR. Should Colstrip #1 ever 
be moved to the 500 kV bus directly, severe SSR problems might 
result.
Electromagnetic Transients Program (EMTP)
A time-domain analysis capability is essential in 
determining the effectiveness of any proposed control action 
particularly for nonlinear systems. The EMTP provides this 
capability with the advantage of having validated, built-in 
models for all of the generation and transmission components. 
The EMTP accomplishes this by integrating a large set of 
differential equations and solving them simultaneously with 
a set of algebraic -- and sometimes nonlinear -- auxiliary
29
or constraint equations. In conventional SSR studies, the 
EMTP can provide valuable information on the magnitude of 
oscillations at a particular frequency and for a particular 
disturbance. None of the other analysis techniques discussed 
here can do the same.
The EMTP is used exclusively in Chapter 5 to carry out 
system and controller simulations.
Eigenvalue Analysis
Eigenvalue analysis can provide more information about 
potential problems than can a frequency scan because the 
eigenvalue analysis includes generator models . One can readily 
obtain the network resonant frequencies (the imaginary part 
of the eigenvalues) from a frequency scan and then compute 
the shaft resonant frequencies as was shown in Figures 6 and 
7 . In addition to all this information, an eigenvalue analysis 
provides the damping information at each resonant frequency.
Many computer programs exist and are widely used that can 
perform these analyses. Eigenvalues can also be obtained 
from an EMTP study as is described in the next section. This 
is the approach taken in this work.
System Identification and Modeling
System identification is important in this research in 
that one of the primary control algorithms to be studied
30
requires the knowledge of a system transfer function from 
input to output. The transfer function we seek is a ratio 
of polynomials in the Laplace complex variable s. The roots 
of the denominator polynomial of this transfer function are 
the eigenvalues of the system as described earlier.
Assuming valid results, the transfer function obtained 
from a system identification routine is a linear model of the 
original system. The nonlinear power system is of extremely 
high order so any model, particularly a linear one, is of a 
reduced order. The identified model is useful in system 
analysis and in designing controllers.
An extension of the Prony signal analysis work completed 
in [41] is used as a system identification routine. More 
information on Prony analysis can be found in [42] through 
[45] . The results of a Prony analysis used to identify a 
transfer function can be put in the form
m R .
G(s)= I - jT, Xi ^ X j V i  * j  (3.1)
i = lS — A;
where R i are the residues of the transfer function G at the 
eigenvalues Xi . For the single input, single output case.
G(s) = 4y(03
4w(0]
(3.2)
where here y(f) is the output response to the input signal u ( t ) .
31
The identification is performed by first giving the system 
an input signal. The input signal should contain components 
of all of the frequencies of interest in the study. The 
systems output response to the signal is observed and recorded. 
Finally, using the known input signal and the observed response 
the identification routine computes a transfer function. The 
eigenvalues of a 12th order model of the MPC system as seen 
from the terminals of Colstrip #3 are shown as x's in Figure
8. The o 's represent the roots of the numerator polynomial 
of G, commonly referred to as zero's.
I
<
300
200 -
100 -
-100 -
-200 -
-300
-1.4 -1.2 -I -0.8 -0.6 -0.4 -0.2
DAMPING
Figure 8. Eigenvalues of the MPC System from Colstrip #3
32
Output Analysis of Pronv Identification Routine
In the application presented by Trudnowski in [41], the 
number of eigenvalues contained in G are wholly determined 
by the number of data points in the output response y(t). The 
number m in (3.1) is chosen so as to optimize the numerical 
robustness of the singular value decomposition (SVD) which 
is an integral part of the Prony package. The SVD is used 
to fit the eigenvalues of the system in a least-squares sense. 
The ordering of the eigenvalues from I to m is somewhat 
arbitrary. It is determined by the order in which the roots 
are taken from the SVD. A difficulty arises in truncating 
the transfer function G to a reasonable order for control 
calculations. One would like to order the eigenvalues so 
that a minimum error exists between the known response y(t) 
and the response predicted by G for a similar input u(t) for 
any given transfer function order n<m.
A first step in ordering the m eigenvalues is commonly 
to group the conjugate eigenvalue pairs together. A real­
valued output response will always present complex eigenvalues 
in conjugate pairs. Following this grouping, Trudnowski uses 
a search technique reported by Hocking and Leslie [46] to 
complete the ordering. Though this technique works well for 
the majority of systems, a major problem was encountered using 
this simple search on some of the systems in this study.
33
To use the Hocking and Leslie search technique, one would 
group the conjugate pole-pairs together and apply the known 
input signal u { t )  to each pole-pair individually. If the pole 
is not complex, u { t )  would be applied to the real pole separately 
from the conjugate poles. The error is tabulated between the 
known output response y ( t )  and the computed output response 
for each pole or pole-pair given the known input. The pole 
or pole-pair having the least error is ranked first. The 
process is repeated for each of the remaining poles or 
pole-pairs using the known input signal on the combination 
of the first k ranked poles along with each of the unranked 
poles or pole-pairs individually. The process can be described 
iteratively as follows:
i = l S  ~  Xi
Ej  =  \ y ( t ) - L u w GjtW + S —X;
E r =  min [£,]
• k< j<m J
’, j  =  k  +  \ , k  +  2 , . . . , m
(3.3)
G*+1W  — GtCs)+
s - X r '
Where the jth eigenvalue is complex, its conjugate, the j+lst 
eigenvalue, must also be included implicitly in the iteration.
The problem in using this type of search technique is 
best described by example. Consider the fourth order system 
described by
34
G(S):
I
- +  -
-I
- +  -
-I
. (3.4).s+ 0 .1+y i50  5 + 0 .1 - y  150 5 + 0.11+7*150 5+0.11-7*150
It is readily seen from the imaginary part of the 
eigenvalues in (3.4) that the oscillators in this system 
operate at the same radian frequency in the s-plane. The 
Sign of the residue of an eigenvalue indicates which direction 
the system response will take due to an external input with 
respect to that eigenvalue. Opposite signs indicate opposite 
directions. With the radian frequency the same and opposite 
sign residues, one can expect some cancellation in the output 
waveform for a net reduction in output magnitude. In fact, 
if the damping in (3.4) were to be equal throughout, the 
output response would be identically zero for any input 
indicating complete cancellation. In this example, the damping 
has been.chosen to be slightly different in the last two terms 
of (3.4) because the Prony routine cannot identify non-distinct 
poles.
This test system was excited by a short series of input 
pulses and simulated at 100 Hertz using a commercial numerical 
package [47] . An output sequence 350 samples long was entered 
into the Prony identification routine. Table I shows the 
partial results of the Prony analysis. The true and estimated 
residues and poles for the pertinent elements of G are shown.
35
Table I. Prony Analysis Results for Example System.
Residues Eigenvalues
Pole # Real Part Imag. Part Real Part Imag. Part
I (true) 1.0000000 0.0000000 -0.1000000 150.00000
(est.) 0.9974675 0.0407073 -0.0999565 149.99980
2 (true) 1.0000000 0.0000000 -0.1000000 -150.00000
(est.) 0.9974675 -0.0407073 -0.0999565 -149.99980
3 (true) -1.0000000 0.0000000 -0.1100000 150.00000
(est.) -0.9974675 -0.0407073 -0.1100044 150.00020
4 (true) -1.0000000 0.0000000 -0.1100000 -150.00000
(est.) -0.9974675 0.0407073 -0.1100044 -150.00020
Since the order of the system is not a priori knowledge 
when performing a Prony analysis, the Prony algorithm 
identifies many extraneous eigenvalues in addition to the 
ones shown in Table I. In this case, some potential eigenvalues 
were discarded in the singular value decomposition and 52 
eigenvalues were listed in the final Prony output. A suitable 
ordering scheme would put the four valid eigenvalues ahead 
of any extraneous ones. The Hocking and Leslie scheme, 
however, ordered the four eigenvalues shown in Table I in 
last place -- behind the 48 other invalid eigenvalues. This 
is because the cancellation due to the presence of similar 
eigenvalues with opposite-sign residues creates an output 
waveform which is generally small in magnitude while the 
response of a valid pole-pair is comparatively larger. Each 
of the valid pole-pairs taken individually create a +34 to
36
+40 dB error between the actual system output and the estimated 
output using that pole-pair. Both pole-pairs taken together 
estimate the actual system extremely well. The error between 
actual and estimated is -260 dB. Figure 9 shows a plot of 
the error as the poles were ordered by the Hocking and Leslie 
search technique.
Response Error vs. System Order
CQ -80 -
l ly-yl l'-100 -
>120 -
-180 -
-200 -
-220 -
-240 -
-260 -
System Order
Figure 9. Errors Between Actual and Estimated Response for
the System in (3.4).
One of the greatest benefits of Prony analysis over other 
identification routines is in the partial-fraction expansion 
form of the program output. Each eigenvalue is isolated from 
the others along with its residue. This property makes for 
easy interpretation of the results. It can generally be said 
that a large residue will elicit a large response from its
37
eigenvalue due to a given input. This is certainly true for 
eigenvalues of similar frequency and damping. One may be 
tempted to order the eigenvalues in a Prony analysis according 
to the magnitude of their residues. This approach would 
certainly solve the problem of the example described by (3.4). 
While the residues of the valid poles have magnitudes of 
approximately one, the residues of the eigenvalues that the 
Hocking and Leslie scheme ranked first had a magnitude of 
IO'15. In fact, the largest residue of the 48 invalid poles 
had a magnitude of IO'14. Although encouraging in this case> 
ordering the eigenvalues according to the magnitude of the 
residues has a distinct disadvantage in that it does nothing 
to try to control the error between the actual and estimated 
outputs.
The method used in this research was a combination of the 
two techniques. It was felt that the additional complexity 
involved in changing the Hocking and Leslie scheme to 
accommodate opposite-sign residues 'would not enhance the 
overall Prony package and may in fact have long-term detri­
mental effects on the robustness of the computer code. Once 
this "patch" was made it would be tempting to use more patches 
to fix future unforeseen problems. With this in mind, the 
Prony code was modified to order the eigenvalues according 
to the magnitude of the residues and separately, according 
to the original Hocking and Leslie scheme. When the user
38
wishes to select an appropriate model order, a computer program 
performs an error analysis on the two files containing the 
separately ordered eigenvalues and choses the model exhibiting 
the least amount of error. This method seems to incorporate 
the benefits of both ordering schemes while not adding sig­
nificantly to the complexity of the code.
It should be noted in closing this chapter that, as always, 
there is no substitute for "engineering judgement" in selecting 
the order of the system model and the eigenvalues that should 
be included in that model. Where there is no knowledge of 
the system to be identified, the Prony identification package 
discussed herein works well. In other cases, one is wise to 
use Prony in conjunction with judgement.
39
CHAPTER 4
\
CONTROLLER DESIGN
Four analysis techniques for determining the existence 
and severity of SSR problems in electric power systems were 
described in Chapter 3. A powerful approach in dealing with 
the task of designing a controller is the last technique 
described -- system identification and modeling. Almost all 
classical control theory, and most modern control theory, 
assumes a known linear "plant" which is to be controlled. 
The plant in the case of these studies is the turbine shaft 
mass-spring system coupled magnetically to the electrical 
network.
An identified model obtained from a Prony analysis is a 
linear model which can be taken to be an accurate representation 
of the plant within a small operating region. A controller 
can be designed to act on the identified model in such a way 
as to increase the damping of the system at the frequencies 
associated with the turbine shaft modes. When the controller 
is implemented on the actual system, it can be expected to 
improve the damping of the system within a certain region
40
where the plant behaves linearly. If this region is very 
large, the controller is said to be "robust" in its operation 
on the plant.
The major goal of this research is to design a robust 
controller to control a resistive brake while meeting certain 
performance criteria. The brake should provide enough damping 
at frequencies within the realm of SSR to allow for the 
reliable use of high-speed reclosing of transmission lines 
emanating from the generator. This is to say that high-speed 
reclosing could be applied without fear of turbine shaft 
damage.
A good starting point in searching for an appropriate 
control law would seem to be straight feedback control. The 
Purdue studies used this approach and showed reasonable damping 
in the turbine shaft following a disturbance [27]. In these 
studies, the deviation from synchronous speed (Acq = CO^-G)0) of 
the generator mass is used as a feedback signal. The author 
best describes the control action in the published results 
of the studies in saying, "... power is dissipated whenever 
the generator speed exceeds synchronous." In other words, 
the brake is turned on whenever Aco>0.
In retrospect, this starting point has emerged as the 
preferred strategy for control of the brake. This report 
will promote the conclusion that the strategy outlined above 
is one of the best possible control algorithms to achieve the
41
damping goal. The Purdue studies give only intuitive evidence 
that suggests straight feedback control on the generator Ato 
would be a good control strategy. This research lays the 
theoretical foundation that appears to validate the premise.
Straight Feedback Control
There are several questions that may be asked regarding 
the use of generator Ato as a feedback signal. Two important 
ones are: "Why does generator Aro work well?" and "Are there 
any other signals that are better?". These questions will 
be addressed in this section and the advantages of using 
generator Ato as a feedback signal will be demonstrated.
Insight into the first question may be gained by using 
an identified model to obtain a root locus plot of the system. 
The open-loop poles for a 12th order model are shown in Figure 
8. A root locus plot extends the results shown in Figure 8 
to include the closed-loop poles of the system. The closed-loop 
poles are the eigenvalues of the closed-loop system. To 
satisfy the design goal, an engineer would like to place the 
closed-loop poles as far into the left half-plane as possible. 
This would provide maximum damping. Figure 10 shows a root 
locus plot of the identified model of the MPC Colstrip test 
system. As in Figure 8, the x's represent the open-loop poles
42
and the o's represent the open-loop zeros. The dots represent 
the location of the closed-loop poles as the feedback gain 
is increased from 0.1 to 1.0.
300
200 -
6  100 -
I .
I
^ -100 -
-200 -
-3001----- 1----- 1----- i-f----1----- 1----- 1-----
- 4 - 3 - 2 - 1 0 1 2 3
DAMPING
Figure 10. Root Locus of the MPC Colstrip 
Test System at Colstrip #3 using Generator 
Ad) as a Feedback Signal.
All of the closed-loop poles move farther into the left 
half-plane as gain is increased indicating improved damping. 
In fact, this feedback is having a dramatic effect on the 
closed-loop poles. An indication of how far the closed-loop 
poles will move for a given change in feedback gain can be 
gained by examining the magnitude of the residues associated 
with a given set of poles. The residues are the Ri' s shown
43
in (3-1) . The generator Ato as a feedback signal provides 
for relatively large residues at the poles corresponding to 
the turbine shaft natural frequencies.
An example of a feedback signal that provides for smaller 
residues at the shaft modes is seen in Figure 11. This figure 
shows a root locus of the same system for the same range of 
gains using generator acceleration as a feedback signal. The 
open-loop poles are in virtually the same location as in 
Figure 10. The fact that the Prony analysis identified the 
same open-loop poles for both cases is an indication that the 
identified models are accurate.
For the moment we ignore the direction in which the 
closed-loop poles are moving and focus attention on the 
distance that the poles move for a given control action. The 
assumption is that with the proper control law, it is possible 
to get the poles moving into the left half-plane. Even with 
this generous assumption, it would be impossible to achieve 
the same amount of damping as can be seen in Figure 10. The 
magnitudes of the residues in the latter case indicate that 
it is more difficult to control the shaft modes using this 
feedback signal.
Another example of testing the various feedback signals 
for suitability in controlling the brake is seen in Figure
1 2 . This case shows the high pressure turbine Aco with the
44
300
200 ■
U 100 -I ..I^
 -100 -
-200-
-3001-------- '------- ' ' .......‘ ------'-------- •-------- 1--------
- 4 - 3 - 2 - 1 0  I 2 3
DAMPING
Figure 11. Root Locus of the MPC Colstrip 
Test System at Colstrip #3 using Generator 
Acceleration as a Feedback Signal.
same range of gains. The HP turbine is on the opposite end 
of the turbine shaft from the generator and is therefore 
readily accessible for speed measurement.
Most of the residues are of large magnitude in this case, 
but it would be nearly impossible to design a control strategy 
that would send all of the closed-loop poles into the left 
half-plane. Thus, it makes little sense to develop a complex 
control strategy when a feedback signal is available which 
does not require complexity as is illustrated in Figure 10.
45
DAMPING
Figure 12. Root Locus of the MPC Colstrip 
Test System at Colstrip #3 using High Pres­
sure Turbine Am as a Feedback Signal.
It should be noted that in all cases, the closed-loop 
poles must eventually end up on the open-loop zeros when the 
feedback gain is increased to infinity. In the three previous 
figures, this means that as gain is increased, the closed-loop 
poles will enter the right half-plane and the system will go 
unstable. One must be careful to remember that the identified 
models used to produce the root locus plots are only good for 
a finite operating range. Hence, as the gain is increased, 
nonlinearities in the system will make the use of root locus 
as an analysis tool unreliable.
46
Filtering the Feedback Signal
It was mentioned in Chapter 3 that one of the modes of a 
turbine shaft mass-spring system involves all of the masses 
on the shaft oscillating in unison with each other. This 
mode has been called an electromechanical mode in some circles, 
or "hunting" mode as in [27] . Since all of the masses on the 
shaft participate in phase with each other for this mode, no 
torques are induced on the shaft. The hunting mode is more 
of a concern in transient stability studies than in transient 
torque studies. This mode is seen as the slow mode which is 
closest to the real axis (zero radians) in Figures 10 through 
12 .
In Figure 10 it can be seen that this mode is well damped 
in the closed-loop. The damping which is added to the hunting 
mode represents wasted energy if the goal is simply to reduce 
transient torques. To this end, it makes sense to let the 
hunting mode maintain its original damping and expend more 
energy damping the other shaft modes. A filter is employed 
for this purpose. The filter acts to remove the hunting mode 
from the generator Aco before it is fed back to the brake. 
Energy saved by not having to damp the hunting mode is expended 
on damping the SSR modes. In Figure 13 a high-pass filter 
has been added in the feedback loop. It can be seen that the
47
filter has very little effect on the higher frequency shaft 
modes, but the hunting mode is effectively stopped from moving 
left.
DAMPING
Figure 13. Root Locus of the MPC Colstrip 
Test System at Colstrip #3 using the Fil­
tered Generator Am  as a Feedback Signal.
The elimination of the hunting mode should be evaluated 
on a case-by-case basis. Figure 10 and Chapter 5 show that 
the hunting mode can be damped with a resistive brake. Many 
different strategies could be employed in the specific design 
of a brake. A design engineer could decide that transient 
stability is a big enough problem to warrant leaving out the 
high-pass filter. This approach would tend to damp the hunting
48
mode along with the torsional modes. Another design might 
filter the feedback signal just until the torsional modes 
achieve a set amount of damping and then switch the filter 
out of the circuit to attack the hunting mode.
Optimal Control
We have seen that the generator Aco may be the best feedback 
signal to use in controlling the brake, but is a straight 
feedback gain the best strategy in using the signal? It is 
common in power system control applications such as exciters 
and PSS units to condition the feedback signal before applying 
it to the controlled input. In addition, optimal control 
theory would suggest that there is a single, unique set of 
control actions which will bring a disturbed output signal 
to zero in a minimum amount of time. In this section, these 
topics are discussed and straight feedback control using the 
generator Aco is again shown to be extremely suitable to achieve 
the design goals.
Conditioning of the feedback signal was discussed in the 
previous two sections. There is no need for complex control 
strategies to be employed in the feedback loop when all of 
the closed-loop poles satisfy the design goal with a simple 
feedback gain. There may be, however, a call for a high-pass 
filter to remove the hunting mode from the feedback signal. 
The hunting mode is well below the lowest SSR frequency and
49
so a second order filter should be adequate. The only strict 
requirement would be to induce a minimum phase error into the 
feedback signal in the SSR range.
The topic of optimality may deserve a more detailed 
explanation. An effort has been made in this research to 
find an optimal control law which would work well with the 
generator Ago as a feedback signal in damping the transient 
torques. A recently developed theory called Generalized 
Predictive Control (GPC) was evaluated as to its usefulness 
in this respect. While GPC researchers make no claims that 
the technique results in the unique optimal control for a 
given circumstance, it can be thought of as a " quasi-optimal1 
control. GPC is formulated for use with digital controllers.
GPC was originally developed by D . W. Clarke of Oxford 
University and his associates in 1985 [48]. Bitmead et al. 
give a more comprehensive discussion of the technique in 
making strong comparisons to linear quadratic control [49]. 
The development of the theory is somewhat similar to that of 
linear quadratic control in that one forms a quadratic cost 
function, J , and attempts to minimize / throughout time. Since 
GPC is a discrete-time algorithm, the cost function takes on 
the general form
N2 n ■ NU
J=  X y (J) +  E  au (k)
J-N1 Zr = I
(4.1)
50
At each successive time step, the objective is to find the 
set of control actions u(i),i = \ ,2 , . . . ,NU that minimize J .  The
control action taken at the next time step is then m (1). The
parameters N1, N2, and NU are somewhat arbitrary, however 
there are some guidelines that can be followed in choosing 
them. Let us assume that N1 = I, JV2=IO, and NU = I . Then
y(i),i = 1,2,..., 10 is simply the pulse response to w(l) predicted
over the next 10 time steps given the initial conditions 
=—n,-n + \,...,§. The control, u, is assumed to be zero for 
all time subsequent to the control horizon NU i.e. 
u(i) = 0,i =2,3,...,°°. Out of all possible choices of u(l), one
would choose the one that minimizes J . As Clarke shows, the 
optimal u can be found by setting the partial derivative of 
/ with respect to ii to zero.
GPC has certain properties that make it appealing in 
situations such as dynamic braking with bang-bang control. 
Suppose one has a discrete-time transfer function fit of a 
system such as can be found using Prony analysis. Then given 
a set of initial conditions, the future values of the output 
can be predicted by iterating the difference equation specified 
by the transfer function. In bang-bang control of the brake, 
there are only two discrete levels of control - the brake is 
either on or off. In the above example where NU = 1, one would 
iterate the difference equation out 10 steps assuming the 
brake is on at m (1) and calculate the value of J .  Then the
51
same thing is done assuming the brake is off at «(1). The 
minimum J of these two possibilities is found and the cor­
responding control is applied. This method is commonly 
referred to in the literature as a tree search. If NU is 
kept fairly small, it can be seen that this would be an 
extremely fast calculation which is important in discrete-time 
control. As NU increases, the number of branches to search 
on the tree increases by Lnu where L is the number of discrete 
levels of control.
Discrete level GPC will theoretically give "more optimal" 
control of the dynamic brake than would straight feedback 
gain. This coupled with the ease of computation in the 
bang-bang case makes GPC look fairly attractive as a control 
strategy.
Sensitivity Problems with Discrete-Level GPC
Despite its inherent advantages, the GPC control strategy 
was found to be ineffective when applied to the dynamic brake 
as specified above. In implementing discrete level GPC, a 
severe lack of robustness with respect to output initial 
conditions was encountered. These problems with sensitivity 
in the algorithm are first illustrated in an example. Following 
the example, an attempt is made to describe the phenomenon 
more quantitatively.
52
The GPC algorithm was implemented on the MPC Colstrip 
model in the EMTP. No positive effects were observed in 
damping the generator A m  for any values of N11 N2, and N U . 
The controller seemed to be acting randomly. After careful 
study, it was found that the 12th order difference equation 
was so sensitive to initial conditions as to make the discrete 
level GPC algorithm inoperable. Ten past values of A m  were 
chosen and the cost function was . calculated for M(O) = O and 
for m (0) = 1. The minimum cost of these two indicated that the 
brake should be off at M(O). A change of only 0.06% in one 
of the past A m  values changed the cost function enough so 
that the minimum cost now indicated that the brake should be 
on at M(0). This behavior was verified on a separate case 
involving only two oscillators and a 5th order system. With 
this kind of sensitivity, GPC as it was implemented originally 
had no chance of controlling the nonlinear system.
As it would have to be implemented in controlling the 
dynamic brake, the GPC algorithm would retrieve past A m  values 
and use these to predict the future values. The sensitivity 
problem surfaces when the noise and nonlinearities in the 
system slightly distort the past A m  values which are used as 
initial conditions in the algorithm. The error in approxi­
mation from the system to the model in this case is enough 
to make the controller ineffective.
53
While the previous example has illustrated the nature of 
the sensitivity problem, it is necessary to study the problem 
analytically in order to fully understand what is happening. 
The discrete-time equivalent of (3.1), assuming a Zero-Order 
Hold (ZOH) on the input, can be written
Given this linearized transfer function, one is faced with 
the task of predicting the future values of the system output. 
Many different formulations of (4.2) can accomplish this task. 
It is initially tempting to write (4.2) as
m — j
^  i-/'7V 1
(4.2)
(4.3)
I - S  4Z-
With
(4.4)
the inverse Z-transform of (4.3) is
m m
y(k) = £ ay Qc - i) + £  ZtiM(£ - i) (4.5)
i = l i = 0
where yQc) is the predicted value of the system output at time 
kTs.
54
Iteration of the difference equation (4.5) to find the pre­
dicted values of the system output given m past values is 
referred to in [50, p,215], [51, p.151], [52, p.474] as the
Direct Form I formulation.
Another formulation seen commonly in control work stems 
from the Direct Form II. We first write (4.2) as
m Qj0Cij +  bj)z-J
£*zoh(z) — b0+ £
I - £ <Z;Z" 
; = i
The state-space representation of (4.6) is
'%# + !)" Cl1 Q1 ... d m - i  a m "%l(t)'
X2Qe + 1) I 0 ... 0 0 X2Qe)
XzQe + 1)
=
0 I ... 0 0 xzQe)
XmQe +1) 0 0 .... I 0 xmQe)
uQc)
yQc') — [(Vi "i ^i) Qjfftz 2^) (Vm + ^m)]
X 1( I c ) '
X1Qc)
XzQc)
+ b0u(k).
(4.6)
(4.7)
xm(k)
Though (4.7) is straightforward and easily implemented 
on a . digital computer, severe numerical problems are 
encountered as the model order, m , is increased. These
55
numerical difficulties are almost wholly due to parameter 
quantization effects in the "a" coefficients. This becomes 
evident through a derivation by Oppenheim [51, p.167] which 
shows the sensitivity of the discrete pole location due to 
small deviations in a coefficient of the denominator polynomial 
of (4.6) . It is shown that the sensitivity problem is 
particularly acute when the poles of the discrete system are 
tightly clustered. Oppenheim concludes by saying,
"[t]hen we can expect the poles of the direct-form 
realization to be quite sensitive to quantization 
errors in the coefficients. Furthermore, it is evident 
that the larger the number of roots, the greater the 
sensitivity."
While Oppenheim speaks of sensitivity in terms of errors 
in the coefficients of polynomials, another measure of sen­
sitivity is commonly applied to matrices. The condition 
number of a matrix A , K(A), indicates the sensitivity of the 
solution vector x to small changes in the entries of the 
coefficient matrix A  and of the vector b in the solution of 
the matrix equation
A x  =  b.
Lancaster [53, p .385] defines the condition number to be
K(A) = IIAII IIA-1I (4.8) 
where |||| denotes any matrix norm. Let us use the spectral 
norm (the matrix 2-norm) to define the condition number. We
know that
56
IIAIIs= max Si
I Si Sn (4.9)
where the subscript s denotes the spectral norm and the script 
Si denotes a singular value of A. We can see that
IIA-1Hs .WA-')
I
n^in(A)
(4.10)
and therefore, at least for invertible matrices,
(4.11)
This tool can now be used to re-examine the twelfth-order 
system shown in Figure 10.
Oppenheim's sensitivity analysis would lead us to believe 
that the formulation in (4.7), the state-space realization 
of Direct Form II, would be a poor implementation of the 
identified model in predicting future output values. An 
evaluation of the condition number of the coefficient matrix 
in (4.7) using the data from Figure 10 yields a K  on the order 
of IO7. This relatively large condition number tends to 
confirm Oppenheim's analysis. Another structure that may be 
used in the implementation of the identified model eliminates 
the numerical problems in predicting future output values 
discussed above. This method is not without its faults, 
however, in that the numerical conditioning problems are 
shifted to another phase of the calculations.
57
The parallel form realization of the identified model is 
better suited for use with the Prony identification routine 
because it implements the system in the partial-fraction 
expansion form characteristic of Prony. Each of the terms 
in the partial fraction expansion is implemented separately 
so that "the error in a given pole is independent of its 
distance from other poles in the system." Oppenheim goes on 
to say, "For this reason, it can be stated that in general 
the ... parallel form [is] to be preferred over the direct 
forms from the viewpoint of parameter quantization." [51, 
p.169]. The conjugate poles can be grouped together in the 
parallel form to avoid complex arithmetic. (4.2) is then 
written
Gzo/z(z) — Co"*" ^
v = il -CCi7-Z-i
mi+T
+ I (V ^ +IV"
6-»i I - CCuZ - CC2fZ-
(4.12)
where In1 is the number of real poles, In2 is the number of 
complex poles, and m —m1 + m2. Now form the sub-matrices
Rj — [ccly] I <j </%,
O2C
O
In2
In1 < L^m1+ -
(4.13a)
(4.136)
58
The coefficient matrix for the state-space implementation of 
the parallel form is then
'A1 0 ... 0
0 A2 ... 0
0 0
IM1+ Y
(4.14)
For equivalent systems, the coefficient matrix (4.14) and 
the coefficient matrix in (4.7) are similar matrices in that 
one can be obtained from the other through an invertible, 
linear transformation. While it is known that the eigenvalues 
of similar matrices are identical, their singular values are 
generally not. The condition number of this matrix for the 
system shown in Figure 10 is 5.8 -- seven orders of magnitude 
less than the coefficient matrix for the direct-form imple­
mentation .
In order to use the parallel form in a GPC application, 
a method must be found to enable us to determine the initial 
conditions of the state variables in their new form. . With 
the direct method, the state variables are directly related 
to the m past output values and past input values. These
59
being known quantities, the prediction algorithm is 
straightforward. With the parallel form, however, the past 
output and input values are related to the initial conditions 
of the state variables by a transformation matrix. If we 
denote the transformation matrix as P , then let
x - P x  (4.15)
then
x(k + 1) =P-1APxik) +P-1BuQc) (4.16)
and
A=P -1AP . (4:17)
To find the initial conditions of the state variables in 
the parallel form, let
X(O)=F-1X(O) . (4.18)
A modification of the method used in Ogata [50, p . 571] 
is used to find the appropriate transformation matrix, F . 
Let
Yjz) _ ^ b u z ^ b ^ z 2 
Uiz) k=i I + CilkZ-1 + au?-2
(4.19)
Sn IS
60
where n=ml2 and m is the system order. This formulation is 
sufficient for the work described in this dissertation since 
all of the modes of interest are oscillatory. . The formulation 
could be easily generalized to include real roots by assuming 
in the form of (4.12) . Now multiplying both sides of (4.19) 
by z~l with / = 1,2, „.,2n - 1 yields
6I* an Z~1 _ 62* +0I^ 6IC
aU y___v.____aW
l+alkz ' + O2kZ 2
Z
(4.20)
where
etc.
61
In matrix notation, (4.19) and (4.20) can be written
Y(z)
CfW
Z-1Y(Z)-Z-1C 1U(Z)
C/(z)
z-2Y(z) - Z-2C 1U(Z) - Z-1C 2U(Z)
Cf(z)
bn
-bzi
bn
2^ian
v a2i y
Z
V
b21
bzidu
a Zl y
b12
-bn
a ZZ
f>21+01l(f>ll )
f  , 622a12 ^
a z\ j < (*22 ^
Now define
[X1(Z)I r z-1 i
Cf(z) I + O11Z-1 + a21z~2
% ) z-2
Cf(Z) I + O11Z'1 + a21z~2
% ) = z-1
CfW I + O12Z-1 + a22z-2
• .
Then from (4.21) and (4.22)
(4.21)
I + anz.1 + a21z 2
l+tinz 1 + a21z 2
X jTa12Z ljra22z 2
(4.22)
62
Y(z) X 1(Z)-
Z-1Y(Z)-Z-1C 1U(Z) % )
z-2Y(z) - Z-2C 1U(Z) - Z-1C 2U(Z)
=  B
Xg(Z)
. .
where @ is the first matrix on the right-hand side of (4.21) . 
(4.23) defines the m states used in the state-space formulation
Xk+i=Axk+Buk
where A is as shown in (4.14) . To illustrate this, the first 
two elements of (4.22) are written as
(I + UnZ-1 + S21Z-X(Z) = Z-1U(Z)
(I + S11Z-1 + S21Z-XzCz) = Z~2U (z) .
Simple substitution yields
X 2(Z) = Z-1X 1(Z)
X 1(Z) = Z-1C-S11X 1(Z) - S21X 2(Z) + U(z))
which correspond to a sub-block of A  as shown in (4.13b) . 
As an aside, the 1B matrix is easily programmed since each row 
builds upon the previous one.
A check of the condition number of P revealed that the 
transformation to parallel form had simply shifted the con­
ditioning problem to another place. The condition number of
A
63
p for the system shown in Figure 10 is on the order of IO11. 
It can be shown that this must be the case when a transformation 
is used to improve the condition of A. From (4.8),
K(A)=Iiaiiiia-iII .
Now (4.17) into (4.8) yields
K(A) = IIPiFiI Ilrt-1PII 
<11Z5 II2II^iI2K(A)
=  [k (P)]2k ( A )  .
It follows directly that
I
(4.24)
(4.24) shows that at least some of the conditioning problem 
is shifted to the transformation matrix when an ill-conditioned 
matrix is transformed to a matrix which is more manageable 
computationally.
There is theoretically no reason to stop at 2» — I in (4.20) . 
The method can be extended to attempt a "least-squares" fit 
of past data to the m initial conditions. As an example.
let
64
Y(z) -X1(Z)-
Z-1Y(Z)-F0(U) % )
Z-2Y(Z)-F1(U) %s(z)
= ®rX2n
I-
--
--
N 4
I I I
__
_
Now fB can be written as
(4,25)
fB = Q tDiSiW 11
where D(Si) is a diagonal matrix of the singular values of tB. 
The Moore-Penrose inverse [53, p.432] of ® is then
B+ = V D 1(Si)Qfl .
Finally, the initial conditions are found by taking the 
Moore-Penrose inverse of (4.24)
x Mt  —  ® 2 nX > ypast * (4.26)
In (4.26) , ypast is the left-hand side of (4.25) .
This extension improves the condition number of tB, but
in practice it was found to weight the past y values too 
heavily. From a control perspective, the controller was not 
allowed to respond quickly to a sudden disturbance because 
only a minor change is immediately seen in the initial con­
ditions due to the weighting.
65
Notwithstanding these problems with the numerical 
implementation of GPC, GPC itself proves to be a powerful 
control strategy. The potential effectiveness of GPC is 
illustrated in the following example. A case was run on the 
linear model in the absence of noise. The control values 
were saved at each time step. These control values were fed 
back into the nonlinear system running in the EMTP for the 
same disturbance. The EMTP output for this case is shown in 
Figures 14 and 15. The series of control actions from the 
separate linear model case worked even in the EMTP, but the 
discrete level GPC could not calculate them given the EMTP 
output data.
Figure 15 shows basically no difference between the 
straight feedback gain strategy and GPC. Although the sen­
sitivity problems with GPC have not yet been worked out, the 
results shown in Figure 15 would lend credence to the claim 
that straight feedback control on the generator Ag) in the 
case of a resistive dynamic brake is in itself "nearly" optimal 
control. For this reason and for reasons discussed in the 
next section, the effort to overcome the sensitivity problem 
of GPC seems to be no more than an academic exercise. Straight 
feedback gain may be as optimal a strategy as can be expected 
for this situation.
66
No Control A
'T  0.3
Q  0.1
Fdbck Gain0 -0 .5
0.2 0.4 0.6
Time (seconds)
Time (seconds)
Figure 14. Uncontrolled Figure 15. Damping of System
Response of System Due to from Feedback Gain and GPC.
Pulse Disturbances.
Sizing the Brake
As related in the introduction, the primary focus of this 
work is in relation to high speed reclosure of circuit breakers 
on series compensated lines. High speed reclosure is a 
technique used to improve system stability by restoring the 
structural integrity of a system as rapidly as possible 
following a temporary fault. A 1979 AEP/MIT paper by Dunlop 
et al. reports experiences reclosing a 765 kV AEP line at 30 
cycles [54] . Previous work suggests that the magnitude of 
damping of transient torques at or near 0.5 seconds after the 
initiating disturbance must be considered in choosing resistor 
values for braking transient torques in a system of this type.
67
Figures 16 and 17 show the generator Aco and torque between 
the generator and the low pressure turbine of the synchronous 
machine shown in the IEEE Second Subsynchronous Benchmark 
(Figure I). The disturbance is a three phase bolted fault 
lasting 4.5 cycles at bus 2. The fault naturally extinguishes 
at 0.0916 seconds with no circuit breaker operation. No 
effort has been made to determine a worst case scenario. The 
above fault scenario was used merely to produce a large torque 
in the generator-LP shaft section. Straight feedback of the 
generator speed signal is used to obtain the damped response 
shown in the plots . The thyristors are gated when the generator 
Am  is greater than zero. The brake size is 240 MW or 40% of 
rated generator MVA.
Generator 
speed dev.
Gen-LP '<—  Uncontrolled
—  Controlled
— Uncontrolled
— Controlled
Time (seconds)Time (seconds)
Figure 16. Second Benchmark Figure 17. Second Benchmark 
Generator Speed Deviation. gen-LP Torque.
68
Studies to date indicate that a major factor in determining 
the size of the brake will be the amount of "dead-time" allowed 
in the reclosure scheme. Dead-time as used in [54] means the 
amount of time which elapses between a circuit breaker trip 
to clear a fault and the subsequent reclose. Less dead-time 
in the reclosure scheme indicates the need for a larger brake. 
In the figures above, a value of 240 MW was chosen for the 
brake to meet a design goal of a maximum of 0.75 p.u. torque 
at 0.5 seconds. Maximum peak to peak torque of 0.70 p.u. 
occurs in the damped curve at 0.6 seconds. Half this value 
is added to the steady-state torque value of 0.38 p.u. to get 
a theoretical maximum of 0.73 p.u. Torque at 0.5 seconds. 
An unsuccessful reclose in the no control case could seriously 
damage the shaft as there is a possibility that the circuit 
could reclose into a 2.28 p.u. Torque creating an intolerably 
high total torque.
Aside from simply satisfying a design goal, the root locus 
plots could be used as a tool to aid in sizing the brake. 
In order to end up on the open-loop zeros in Figure 10, the 
closed-loop poles must change direction at some point and 
travel to the right half-plane. The value of gain for the 
point at which the closed-loop poles change direction rep­
resents the maximum possible damping for the system. To use 
the root loci as a tool in determining optimal brake size, 
an engineer would perform a Prony analysis on a system with
69
an arbitrary brake size —  say 100 MW. The identified model 
is used to plot a root locus. The root locus is examined to 
determine the gain at which the poles change direction. This 
gain is a multiplier on the control action. If the gain 
should turn out to be 3.0, the controller would be calling 
for three times a 100 MW brake or 300 MW. One would have to 
remember in using this approach that the identified model is 
only valid for a small operating region. The brake size 
determined using this method would have to be confirmed using 
other means.
We have shown that the generator Aco is an excellent 
feedback signal to use in controlling the dynamic brake. We 
have also implied that straight feedback gain on this signal 
is nearly optimal. This section on sizing the brake lends 
more credibility to the claim that complex controllers or 
optimal digital control is not needed in this case. Simply 
employing a larger brake will have the same effect as 
implementing a better control law if such a law exists and 
if economics permit a larger brake.
70
CHAPTER 5
SIMULATION RESULTS
The simulation results in this section were performed 
using EPRI's Electromagnetic Transients Program (EMTP) Version 
2.0. Synchronous machines in EMTP are modeled in a rotating 
orthogonal coordinate system using Park's equations and the 
associated transformation. The mechanical mass-spring system 
is modeled separately by Newton's laws for rotating bodies. 
No governor model has been included in the simulations and 
so the EMTP assumes a constant mechanical input power.
These results were obtained using the straight feedback 
control strategy discussed in this section and listed in 
Appendix C. The brake is switched in a bang-bang mode rather 
than using a sophisticated firing angle thyristor gate control 
which could simulate a variable resistance and provide smooth 
control. In addition to simpler operation, bang-bang control 
should provide faster damping. To implement the algorithm, 
a gain is set for the feedback loop and a dead-zone is 
established where the brake will not be turned on. The 
dead-zone serves the function of disabling the brake when
71
there is no major disturbance present so that small fluc­
tuations in generator Aco do not activate the brake. When 
the feedback signal exceeds the dead-zone limit in the positive 
direction, the brake thyristors are gated. Dalpaiz [55] shows 
that the proposed feedback signal, Ato, is accessible on a 
typical generator. In this paper, the generator Aco is 
monitored for subsynchronous oscillations using a PC.
The results in this section have been chosen to illustrate 
certain points with respect to the application of the brake 
which have been mentioned in the previous sections. The 
effects, or lack thereof, of filtering, brake size, control 
algorithms, and thyristor types are seen in the plots. A 
discussion of the effects of dynamic braking on nearby machines 
is also included in this section.
Effects of Filtering
To study the effects of a filtered Aco as discussed in 
Chapter 4, a simple second order high-pass filter was designed 
with a cutoff frequency of 2 Hz. In order to effectively 
damp the torsional modes, it is imperative that the filter 
have minimum phase error at 20 Hz and above -- the range of 
subsynchronous resonance and shaft torsional frequencies. 
With a decade between the cutoff frequency and the passband, 
a second order filter can meet these specifications. The
72
s-plane transfer function was converted to discrete-time using 
a bilinear transformation and implemented in the controller 
subroutine shown in Figure 5.
The figures in this chapter were obtained from EMTP 
simulations of the MPC Colstrip model described in Chapter 
2 . In the first set of cases to be discussed, Colstrip 3 and 
4 are lumped together to form a conglomerate 1638 MVA machine. 
The brake is operating at the terminals of this machine. 
Colstrip I and 2 are operating separately into the 230 kV bus 
CS230 (Figure 2). A three phase bolted fault occurs at bus 
BGFFl at 0.22 seconds. The fault is cleared at buses BGOOl 
and GBSWl 4.7 cycles later. The circuit breakers at BGOOl 
and GBSWl then reclose into the same fault at 1.112 seconds. 
This gives a total dead-time of 0.814 seconds. The fault 
clears itself 2 cycles later with no further breaker opera­
tions . The disturbance was chosen to produce large torques 
in the turbine shaft. Reclosing into a three phase fault 
would be expected to be one of the most severe disturbances. 
The brake is rated at 200 MW or 12 % of the conglomerate rated 
MVA.
Figure 18 shows the uncontrolled conglomerate machine Ago 
for this disturbance while Figure 19 shows the controlled but 
unfiltered response superimposed on the filtered, controlled 
response. Of particular interest is the sharp increase in 
amplitude of the torsional modes at the time of reclosure.
73
It is this torsional impact that is of major concern in these 
studies. Both controlled responses show a large amount of 
damping over the uncontrolled case in the transient modes at 
I second - just before reclosing.
Generator
speed dev.
CS #3&4 
conglomerate 
speed dev.
— Unfiltered
— Filtered
Time (seconds)Time (seconds)
Figure 18. Uncontrolled Gen­
erator Act) for Three Phase 
Fault With Reclose.
Figure 19. Filtered and 
Unfiltered Controlled 
response for Case in Figure
18.
Figures 20 and 21 show the effects of filtering on the 
control action. The top plot in each figure shows the generator 
Ato as seen in Figure 19 while the bottom half shows the A 
phase brake current. The filtered brake gives more uniform 
action on the torsional modes with less of the long-term 
operation seen between 0.2 and 0.6 seconds in Figure 20.
Figure 22 shows the torque on the LP1-LP2 shaft section 
for both the filtered and unfiltered control. Here the 
benefits of the filter are evident. The plot shows a 36%
74
O 0.2 0.4 0.6 0.6 I 1.2 1.4 1.6 16  2 u °  1
Time (seconds) Time (seconds,
Figure 20. Generator Speed Figure 21. Generator Speed 
Deviation and Brake Current Deviation and Brake Current 
for Unfiltered Controller. for High-Pass Filtered Con­
troller .
reduction in total torque at 1.2 seconds through filtering 
compared to the unfiltered control. Damage to turbine- 
generator shafts due to excessive torque is known to be 
exponential in nature so a 36% reduction in torque could 
correlate to a much larger percent increase in shaft life. 
Moreover, the filtered signal control case is down 57% from 
the uncontrolled torque. The 57% figure may be enough to 
guarantee the safety of the shaft should a high-speed reclosure 
scheme be considered by a utility.
The filtering discussion in Chapter 4 and the results 
shown in Figures 18 through 22 demonstrate the effectiveness 
of employing a high-pass filter on the feedback signal. The 
amount of filtering and the strategies used may be a matter
75
2  is :
<  6 -
—- Uncontrolled
Time (seconds)
Figure 22. LP1-LP2 Torque for Disturbance 
in Figure 18 Showing Effect of Filtering.
of engineering judgement in cases where a secondary function 
of the brake is to improve transient stability. High-pass 
filtering is used on all of the following plots and those 
seen in Chapter 6 unless otherwise specifically stated.
Results with a 200 MW Brake
Two additional torque plots are shown below for the same 
disturbance. Figure 23 shows the HP-IP torque for the 
uncontrolled response superimposed on the controlled. The 
LP2-Gen torque for the same case is shown in Figure 24. 
Special attention is called to the torsional damping at 
approximately 1.1 seconds. The damping at this time -- just
76
before reclosing -- should be used as a measure of the 
effectiveness of the controller since this value can be used 
to approximate the maximum possible torque at reclosing. 
Simulation may not show the maximum torque if the oscillations 
are out of phase with the response due to the reclosing.
0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
Time (seconds)
Figure 23. HP-IP Torque for Disturbance in
Figure 18.
Damping is seen to have improved in all cases, although 
perhaps not as dramatically as is seen in Figure 22. One 
reason for this is that this particular disturbance was "tuned" 
to create large torques in the LP1-LP2 shaft section. An 
evaluation of the eigenvectors associated with the mass-spring 
system has shown this section and the LP2-Gen section to be 
the two most critical for the Colstrip #3 and #4 machines
77
torque here
>< 5 -
i iA JA
CD 3 -
— Uncontrolled 
Controlled
Time (seconds)
Figure 24. LP2-Gen Torque for Disturbance 
in Figure 18.
modeled in these simulations. The root locus plots of Chapter 
4 and further simulations show that the brake would also be 
effective in damping the other shaft modes.
Effects of Brake Size
Some factors in determining the size of the brake were 
discussed in Chapter 4. The following plots were chosen to 
illustrate some of the benefits and drawbacks of using larger 
brakes in attempting to damp torsional modes.
Figure 10 shows that the closed-loop system is non-minimum 
phase i.e. there are zeros in the right half-plane. Arbitrarily 
increasing the size of the brake will eventually force the
78
closed-loop poles to change direction and reduce the damping 
of the system. A reasonable approximation to the point at 
which the poles in Figure 10 change direction is at a feedback 
gain of 5.2. The root locus was drawn for a 200 MW brake. 
From this information a guess might be made as to the optimal 
size of the brake. To make this guess, two major assumptions 
must be made. The system must remain at an operating point 
which is close to the one at which the Prony analysis was 
made so linearity is maintained. Also, one must assume that 
the bang-bang action of the brake is a close approximation 
to the continuous control of the root locus. These being 
stated, a 1,040 MW brake may seem to be a reasonable choice 
from the standpoint of maximum damping ability.
Mainly because of the bang-bang restriction, the 1,040 
MW estimation is too large. Damping is seen to have decreased 
from that of brakes with slightly smaller ratings and the 
large device starts to create some system disturbances of its 
own. Figures 25 and 26 show the LP1-LP2 torque and the HP-IP 
torque for the disturbance described in the previous section 
using an 800 MW brake for damping. In Figure 25, the 800 MW 
brake is shown on top of the 200 MW brake and the uncontrolled 
response. Figure 26 shows only the 800 MW brake and the 200 
MW brake. 800 MW is 50% of the machine rated MVA.
79
0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
Time (seconds)
Figure 25. LP1-LP2 Torque for Disturbance 
in Figure 18 Using 800 MW Brake.
In Figure 25, a large increase in damping can be seen 
roughly between 0.4 seconds and 0.9 seconds. After this time, 
the damping achieved using the 200 MW brake almost equals 
that of the 800 MW brake. This would seem to be a case of 
the large brake creating a mild disturbance by its own operation 
after the amplitude of the feedback signal has been reduced 
a considerable amount. Figure 25 shows that there is no need 
for a larger brake given the amount of dead-time used in this 
reclosing scheme. The use of a faster reclosing scheme may 
warrant the consideration of a brake larger than 200 MW.
Figure 26 tells a different story. There is no indication 
that the larger brake has improved the damping of this faster
80
2.4
2.2
2
CO'1.8 
O  1.6T—
.4
CO 1 2
S  1
®  0.8
o.6
O  0.4 
%:0 2
-0.2
-0.4
-0.6
Figure 26. HP-IP Torque for Disturbance in 
Figure 18 Using 800 MW Brake.
—  200 MW Brake 
800 MW Brake
Time (seconds)
mode. Further studies show that with the given control 
strategy, maximum damping is achieved for this mode with a 
500 MW brake. This may still be larger than is needed given 
the reclosing scheme employed.
Damping Electromechanical Modes
The root locus of Figure 13 illustrates the usefulness 
of filtering the feedback signal to direct more energy into 
the damping of the torsional modes. An undesirable effect 
of the filter used here is the apparent decrease in damping 
of the electromechanical mode. The reduction in damping, 
however, may not be as severe as indicated in Figure 13. If
81
it were, there would certainly be cause for concern. Simu­
lations bear out that there is little detrimental effect on 
the electromechanical modes for low power brakes of the kind 
which would probably be installed specifically to damp 
transient torques. These brakes would be small in size and 
would be active for a very short period of time with respect 
to the frequency ranges of interest in transient stability 
studies. For larger brakes that are meant to deal with the 
stability problem as well as transient torques, the controller 
would be designed so as to provide damping to all of these 
modes.
Figure 27 shows the generator Aco for two cases. In the 
first case, high-pass filtering is employed throughout the 
simulation. Damping of the electromechanical mode is indeed 
decreased with this large brake in operation. In the second 
trace on this plot, the filter was taken out of the feedback 
loop when the filtered signal was adequately damped. This 
allowed the brake controller to act on the electromechanical 
mode with dramatic results.
In damping first the torsional modes and then the 
electromechanical mode, the dynamic brake is seen to be 
extremely flexible in its applications. Designs specific to 
an installation have a wide range of possibilities which need 
to be considered on a case-by-case basis.
82
LP Filter Remaining
o' 2 ■
LP Filter R em oved
Time (seconds)
Figure 27. Generator Aco Signals Showing the 
Electromechanical Mode Damping.
Thyristor TVoes
A question which has arisen frequently centers around the 
type of thyristor needed to obtain effective control using 
the scheme outlined above. SCR' s are commonly used as switches 
in power system applications. This makes them attractive for 
use in switching a dynamic brake, but they have the drawback 
that once gated, they will continue to conduct current until 
they are line-commutated which must happen at least every 8 
milliseconds. Gate Turn-Off thyristors or GTO's are less 
familiar but they have the advantage that they can be commutated
83
at any time. This feature is a must for truly optimal control. 
In this case, the brake cannot continue to conduct after the 
controller has given the "off" signal.
Figure 28 shows the generator Ato for two signals in a 
case for which a 200 MW brake is used to damp the disturbance 
mentioned for Figure 18. One of these signals is the system 
response where SCR's were used to switch the brake and one 
for GTO's .
SCR’s vs. GTO s
T  1 ■
.... GTO
— SCR
Time (seconds)
Figure 28. Generator Ato Signals Showing the 
Effects of Thyristor Type.
There is little difference in damping between the two 
signals indicating that SCR's are sufficient in this appli­
cation. The main reason for the success of SCR's is that the 
time lag associated with the commutation of these devices is 
small compared to most of the frequencies being damped. The
84
SCR's are being commutated at 60 Hertz while most of the 
torsional frequencies are below 30 Hertz. One may expect to 
see an advantage in using GTO7s if a high torsional frequency 
is expected to be lightly damped. In addition, because of 
the large amount of energy being expended in the time between 
"off" signal and commutation, large brakes may warrant the 
use of GTO's.
Effects of Braking On Nearbv Machines
The use of a braking resistor at the Colstrip #3 con­
glomerate machine was seen in the simulations as having very 
little effect on the response of other nearby machines. In 
the case of the MPC Colstrip model, two smaller machines feed 
the same bus as does the Colstrip #3 machine where the brake 
was located. The response from these machines was closely 
watched for signs of reduced damping in either the torsional 
modes or the electromechanical modes. No significant effects 
were seen in either case except that when the brake was used 
to damp electromechanical oscillations subsequent to its 
primary purpose, an improvement in the damping of the elec­
tromechanical modes at Colstrip #1 and #2 were seen also. 
Figures 2 9 and 3 0 show the generator Ag) of Colstrip #1 for 
the same disturbance that has been previously cited. In 
Figure 29, the uncontrolled disturbance is shown as a ref-
85
erence. Figure 30 shows the same response when a 200 MW brake 
and a 800 MW brake are used to control the torsional 
oscillations on Colstrip #3.
... 200 M W  Brake 
“  800 M W  Brake
No Control
Time (seconds) Time (seconds)
Figure 29. Colstrip #1 Figure 30. Colstrip #lGen-
Uncontrolled Generator Speed erator Speed Deviation for 
Deviation. 200 MW and 800 MW Brake.
It can be seen from these figures that the application 
of a brake acting on one machine has little effect on other 
nearby machines. In the case of the MPC Colstrip model, this 
is because the non-controlled machines do not have any shaft 
modes that are near to the modes of the Colstrip #3 conglomerate 
machine. The braking action on the Colstrip #3 modes does 
not excite the modes of Colstrip #1 or #2. If the modes of 
a nearby machine fall close to those of the machine being 
controlled, one could expect to see a more significant effect 
of the brake on the uncontrolled machine.
86
In this chapter we have shown some standard configurations, 
and attempted to illustrate some of the points which have 
been brought up in the previous sections. Chapter 6 shows 
the simulation results for some special configurations and 
operating conditions which are of interest in this study.
87
CHAPTER 6
EFFECTS OF FAULT TYPE, POWER FACTOR AND BRAKE ORGANIZATION
It is well known that unbalanced network faults create a 
negative sequence component of current which can be thought 
of in the phasor domain as a counter-rotating current. For 
simplicity, a single-line-to-ground (SLG) fault will be 
considered in this discussion although equally valid arguments 
can be made for other unbalanced fault types. The negative 
sequence current exists in the stator of the generator both 
during the fault and during the clearing action if single-line 
clearing is employed.
It is logical to assume that the counter-rotating currents 
flowing in the stator of the generator as described above 
will set up a second harmonic torque in the generator rotor 
as the magnetic fields associated with each will cross twice 
per revolution of the rotor. It has been suggested that this 
second harmonic component may have some uses in the feedback 
control of the dynamic brake to damp transient torques. 
Overall damping of a machine, however, is said to increase 
when negative sequence currents are present [56] . With these 
preliminaries in mind, an EMTP case was set up to simulate
88
a SLG fault with single-line reclosing for analysis. The 
brake was not included in this analysis. Fast Fourier 
transforms were performed on the generator Am signal and all 
of the applicable torques of the single synchronous machine 
in the simulation. The Montana Power Company Colstrip test 
system was used as the model. Figures 31 through 35 show the 
results of these FFT's. A small 120 Hertz component of 
generator speed deviation is seen in Figure 31 but no sig­
nificant components are seen in the torque plots.
Further examination of the FFT's reveals that the sub- 
synchronous frequencies corresponding to the turbine shaft 
modes are prominent and higher frequencies are attenuated 
greatly. This confirms results and discussions on the topic 
found in conventional texts and in the open literature. In 
fact, the supersynchronous component of network resonances 
set up by such components as series capacitors are not con­
sidered to be a problem in SSR studies. This higher frequency 
component is generally not exciting the turbine shaft at one 
of the shaft's natural frequencies, which dominate the FFT 
and so that component is generally neglected. We could find 
no advantage to utilizing the second harmonic frequency in 
the control algorithm.
89
Generator velocity
A m .
Figure 32. FFT of HP-IP 
Torque.
IP-LPl Torque LP1-LP2 Toique
Figure 33. FFT of IP-LPl 
Torque.
Figure 34. FFT of LP1-LP2 
Torque.
The FFT's confirm what is seen from system transfer 
functions obtained from Prony analysis. Both show an 
overwhelming dominance of the turbine-generator shaft modes 
in favor of any network components that the dynamic brake may 
see. This would suggest that with the given feedback signal, 
the brake devotes almost all of the available energy to damping
90
Figure 35. FFT of LP2-Gen 
Torque.
the particular shaft resonant modes regardless of the network 
configuration at any given moment in time. In other words, 
the controller does an effective job of damping the shaft at 
the frequencies of interest through a disturbance of any type 
- be it in the electrical network or even a mechanical 
disturbance.
Figures 36 through 38 compare the uncontrolled response 
to the controlled response of the Colstrip model when subjected 
to a SLG fault near to the synchronous machine. The fault 
occurs at 0.22 seconds and is cleared by a single line opening 
at 0.30 seconds. An unsuccessful single-line reclose occurs 
at 1.11 seconds and the fault then extinguishes itself at 
1.16 seconds with no further circuit breaker operation. The 
generator Ato (feedback) signal is high-pass filtered to block 
the electromechanical modes of the generator. A "dead-zone"
91
is employed in the control algorithm to prevent the brake 
from operating unless there is a significant disturbance in 
the system. The dead-zone encompasses a region where the 
filtered signal is approximately ± 0.01 radians per second 
from synchronous speed. It can be seen that the brake provides 
excellent damping of the transient torque over the uncontrolled 
case.
1.2 
1
0.8 
0.6 
0.4
2?-0.2 
-0.4 
-0.6 
-0.8 
-I
-1.2
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
Time
Figure 36. Generator Aco Signals Showing 
Damping of SLG Fault.
Generator speed deviation
No control
Controlled
It should be noted that similar results were obtained 
from DLG faults which also contain negative sequence currents. 
As a point of reference, the SLG fault created a disturbance 
in the generator speed deviation signal which amounted to 
approximately one third of the magnitude of the same signal
92
when subjected to a three phase fault at the same bus. The 
damping ratio appears to be similar in both cases, but the 
small initial magnitude of the SLG fault allows for a quicker 
return to steady-state than does the three phase fault.
LP1-LP2 Torque
No control 
- Controlled4.5 -
1.5 -
Time
Figure 37. LP1-LP2 Torque Showing Damping 
of SLG Fault.
The dominance of the shaft torsional modes as seen from 
the generator Aco signal in the FFT plots can also be used to 
explain some of the robustness properties of the control 
algorithm with respect to other system conditions. Figures 
36 through 38 leave little doubt that the controller is robust 
with respect to fault type. Additional simulations involving 
the system at different steady-state operating points and for 
varying generator power factors have resulted in comparable
93
LP2-Gen Torque
4.5 -
•• No control 
Controlled
Time
Figure 38. LP2-Gen Torque Showing Damping 
of SLG Fault.
results to those which have already been illustrated. The 
choice of generator Aco as a feedback signal has again proven 
to be a wise one. Network configurations, fault types, 
generator loading, and generator power factor are not 
parameters which can significantly affect the spectrum of 
this signal. This property is what makes the control algorithm 
robust through a wide range of conditions.
Several studies were also conducted as part of this project 
to determine some options for brake configurations at a 
generation facility. Economic considerations may warrant the 
use of only a single brake at a given facility if a control 
algorithm can be found which will damp all of the machines
94
at that location. A study involving multiple brakes at a 
single facility and a study for a single brake for multiple 
machines are described in the remainder of this section.
Multiple Brakes at a Single Location
Optimal braking for any given machine can only be achieved 
by the use of a dedicated brake for that machine. These 
studies have shown, however, that sets of identical machines 
which are electrically close to one another can be controlled 
effectively from a single brake. The damping seen for these 
machines closely approximates that of a single brake for a 
single machine. Even though the identical machines may be 
loaded differently at any given time, the shafts exhibit the 
same resonances and can be expected to be in phase in response 
to a network disturbance. Only the damping may differ due 
to loading conditions. This analysis is confirmed in [57].
Figures 39 and 40 show the damping of the generator Aco 
signals for two non-similar machines for a case in which two 
dynamic brakes have been used for damping. One of the brakes 
is a 200 MW brake connected to the 2 6 kV CS2 61 bus (see Figure 
2) at Colstrip on the MPC model. 200 MW again represents 16 
% of the conglomerate machine rated MVA. The other brake is 
located on the low voltage side of a transformer connected 
to the 23 0 kV CS23 0 bus. This brake operates at 22 kV and 
is intended to damp the transient torques in both the Colstrip
95
#1 and #2 machines. These are identical machines which are 
both rated at 377 MVA. The second brake is rated at 120 MW 
which is 16 % of the Colstrip #1 and #2 combined rating.
The disturbance is similar to that described in Chapter 
5 and the control algorithm for each separate brake is also 
the same as that discussed in Chapter 5. The Colstrip #3 
conglomerate machine Aco is fed back to control the first brake 
and the Colstrip #2 Ato is used to control the second brake.
— Uncontrolled 
Controlled
Time (seconds)
Figure 39. Colstrip #3 Ago Signals Showing 
Damping for Multiple Brakes at a Single 
Location.
Figures 41 and 42 show the LP1-LP2 torque on the Colstrip 
#3 conglomerate machine and the LP-Gen torque for Colstrip 
#1. The damping for the Colstrip #2 machine is analogous to 
that of Colstrip #1.
96
6 
5 
4
P
I:
I -
i n .2  
|-3 
-4 
-5 
-6
Figure 40. Colstrip #1 Aco Signals Showing 
Damping for Multiple Brakes at a Single
Location.
Alkt. ,I,rtiii
....[Si
Controlled ' 11
Ilpiwj(Hi" 1
0 0.2 0.4 0.6 0.8 I 1.2 1.4 1.6 1.8 2
Time (seconds)
—- Uncontrolled 
— Controlled
Time (seconds)
Figure 41. Colstrip #3 LP1-LP2 Torque Show­
ing Damping for Multiple Brakes at a Single
Location.
97
2.5 ■
a  0.5 -
c 0
— Uncontrolled 
Controlled-1.5 -
Time (seconds)
Figure 42. Colstrip #1 LP-Gen Torque Show­
ing Damping for Multiple Brakes at a Single
Location.
The results shown above seem to be reasonable given that 
the Chapter 5 results showed no effect on the damping of the 
smaller machines using a brake on the larger machines. Special 
precautions may have to be taken if a resonant mode of one 
machine matches that of another machine but the machines are 
otherwise dissimilar.
A Single Brake For Multiple Machines
To complete the study of the various brake configuration, 
a single brake was placed on the low voltage side of a 
transformer at the 500 kV CS500 bus. The generator Ato signals 
for both the Colstrip #3 conglomerate machine and the Colstrip
98
#2 machine were fed into a summing junction and then into the 
controller filter. This simple strategy would have to be 
re-evaluated if some of the modes of one machines fell close 
in frequency to those of another machine. In this case, the 
simple summer may be unpredictable. Components which are in 
phase will tend to activate the brake for added damping. 
Weighting factors could be added to one signal should engi­
neering considerations dictate that one machine should receive 
more damping than others.
A 320 MW brake was used representing 16 % of the total 
rated MVA of all machines at the facility. Figures 43 and 
44 show the generator Ato for Colstrip #3 and Colstrip #1
— Uncontrolled 
Controlled
Time (seconds)
Figure 43. Colstrip #3 Ago Signals Showing 
Damping for a Single Brake on Multiple 
Machines.
99
respectively. The damping seen in these plots is slightly 
less than that seen for multiple brakes which can be expected 
with this strategy. As is consistently the case, the torque 
plots follow the damping seen in the speed plots and so are 
not shown here.
.... Uncontrolled 
Controlled
® 3 ■
Time (seconds)
Figure 44. Colstrip #1 Aco Signals Showing 
Damping for a Single Brake on Multiple 
Machines.
This section has highlighted some of the robustness 
properties of the brake as it is proposed. The control 
algorithm and feedback signal are seen to adapt well to 
numerous different situations including operating points, 
generator power factor, and fault types. It was suggested 
that this is because the spectrum of the feedback signal is
100
dominated by the modes of the turbine shaft mass ^ spring system 
which are constant and thus can be very accurately represented 
by linear models.
101
CHAPTER 7
CONCLUSIONS AND FUTURE WORK
This report has presented discussion and results relating 
to the application of a dynamic resistive brake to damp 
torsional oscillations in a synchronous generator. A computer 
model of a dynamic brake was applied to the EMTP models of 
two separate test systems in a variety of different conditions . 
Feasibility of the scheme was demonstrated by time-domain 
plots showing significant damping of the turbine shaft tor­
sional modes over the uncontrolled case. Some of the benefits' 
of this research and conclusions drawn are summarized in this 
section.
Dynamic braking of the type employed in this study would 
be particularly appealing in situations where large turbo­
generators are served by series-compensated lines. It is 
usually the case in these situations that the generators are 
operating near to their stability limits, and therefore fast 
reclosing of the series-compensated lines is desirable. 
Currently, a major concern with fast reclosing is torsional 
damage to the turbine-generator shaft. The scheme developed
102
in this thesis will improve the damping of the turbine- 
generator shaft transient torques providing a larger safety 
margin for utility engineers contemplating fast reclosing.
The reasons this work should prove to be beneficial in 
future research involving dynamic braking, transient torque 
studies, and SSR studies follow.
• Two detailed system models exhibiting moderate to 
severe transient torques have been developed and 
implemented in the EMTP. One model is a standard 
of the IEEE and may be used to verify results from 
research conducted at various institutions. The 
other model represents a large system with multiple 
machines and is a realistic representation of an 
actual system.
• An EMTP model of a resistive brake was developed 
and tested. The controller for the brake was made 
to be extremely flexible by allowing the use of 
a user-written subroutine interface with the EMTP.
This subroutine receives EMTP variables, makes a 
decision on control action, and transmits the 
control action to the brake thyristors which are 
modeled in the EMTP.
• Several traditional SSR analysis techniques were 
discussed. System identification using Prony's 
method was introduced and proved to be particularly
103
useful in developing control strategies. In 
addition to providing information on the eigen­
values of a system as does the more conventional 
Eigenvalue Analysis technique, Prony's method 
provides the user with the residues associated 
with those eigenvalues. The residues and 
eigenvalues can be combined to form a transfer 
function that represents the system as seen from 
the brake.
e Two schemes for ordering the eigenvalues in the 
partial-fraction expansion form of the transfer 
function which is characteristic of the output of 
a Prony analysis were shown. Potential problems 
with these schemes were identified and a compromise 
solution was proposed.
• Sensitivity problems in implementing discrete- 
level GPC on high-order systems were identified. 
The discussion and analysis of these sensitivity 
problems should prove to be beneficial in future 
work relating to optimal control and the use of 
high-order models in control engineering.
« Some techniques for determining appropriate 
feedback signals and control strategies were 
discussed. Root locus plots were shown to be 
helpful in determining the distance that an
104
eigenvalue can be moved using a particular feedback 
signal. This property was related to the magnitude 
of the residue of that eigenvalue. The control 
strategy would have to be one that forces the 
eigenvalue to move in the correct direction.
Some of the conclusions that can be drawn from this research 
are as follows:
• The speed deviation from synchronous of the 
generator on which the brake is acting proves to 
be a superior feedback signal for use in damping 
torsional oscillations of the turbine shaft. This 
signal is dominated by the spectrum of the 
mechanical modes of the shaft. The modes of the 
electrical network are seen to have small residues 
and are therefore weakly represented in the 
generator Ago.
• A straightforward, easily implemented control 
strategy provides excellent damping of the shaft 
torsional modes. This straight feedback control 
was shown to be "nearly optimal" through the use 
of root locus plots and time-domain simulations.
• The search for a truly optimal control law may not 
be worth the effort. The added complexity of 
control may make the algorithm lack robustness. 
Furthermore, should additional damping be required
105
it would be simpler to simply increase the brake 
size rather than attempt to make the brake only 
marginally more efficient.
• A high-pass filter acting on the feedback signal 
can provide more damping energy to the torsional 
modes at the expense of the electromechanical modes 
of the machine. The net result is improved damping 
of the transient torques in the short time frame 
that is considered for high-speed reclosing 
applications.
• The high-pass filter mentioned above may tend to 
reduce the damping of the electromechanical mode. 
This phenomenon may be of little concern given 
that the brake is only intended to operate for a 
few seconds. The brake can either be turned off 
at this time or the filter can be removed from the 
feedback circuit allowing the brake to damp the 
electromechanical modes. Both strategies were 
shown to be acceptable.
<• The straight feedback control algorithm is robust 
with respect to a large range of operating con­
ditions, fault types, generator power factors, and 
brake configurations. An explanation for this 
property is that since the feedback signal is 
dominated by the shaft modes, configurations and
106
anomalies in the electrical network do not sig­
nificantly affect the control action. The shaft 
mass-spring system is linear and time-invariant 
which makes analysis relatively simple.
• For all cases tested, SCR's proved to be sufficient 
in solid-state switches for the dynamic brake. 
No need for more expensive GTO's was observed. 
The delay in commutation of the SCR's will be a 
maximum of 8 milliseconds. This delay does not 
severely affect the control action when damping 
shaft modes which are normally slow.
• A brake on the order of 12% of the rated generator 
MVA should be adequate in damping transient torques 
in conjunction with high-speed reclosing. Larger 
brakes may be more desirable if a secondary 
function of the brake is to damp electromechanical 
oscillations in the machine.
• Several brakes may be used at the same facility 
to damp different machines, however there is no 
need for multiple brakes to damp two identical 
machines which feed the same bus. Their shaft 
modes are also identical and they oscillate 
together when subjected to the same network 
disturbance.
107
• With modifications to the control algorithm, one 
larger brake may be used to damp oscillations in 
more than one non-identical machine feeding the 
same electrical bus. Problems may occur in this 
configuration when two non-identical machines have 
one or more nearly identical shaft modes . Analysis 
of shaft modal frequencies is simple and this 
situation should be easily recognizable should 
this design be considered by a utility.
Future work in optimal control and discrete-level GPC 
should focus on formulating the problem so as to minimize the 
discretization error for high-order systems. The Prony 
algorithm should be generalized to accommodate a feed-forward 
term, multiple eigenvalues, and a residue algorithm that can 
use data collected during the probing phase of the identi­
fication.
Finally, economic studies comparing the cost of dynamic 
braking to the cost of lost generation that may be avoided 
by the use of a braking resistor are needed. These studies 
should be completed before more detailed studies regarding 
the dynamics of the brake are pursued. Loss of generation 
due to stability problems resulting from the absence of a 
high-speed reclosing scheme is one form that may be avoided 
through the use of a dynamic brake. SSR relays are now 
commonplace in steam-turbine generation facilities. It may
108
be possible to reduce the sensitivity of these relays when 
a dynamic brake is in use. This would reduce the number of 
incidents of unnecessary trips at some facilities.
' Along with showing the feasibility of dynamic braking, 
this dissertation gives enough background and analysis to 
obtain a good estimation of the cost of implementing dynamic 
braking at a generation facility. The cost of lost generation 
would have to be determined by the interested utility.
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116
APPENDICES
117
APPENDIX A
IEEE SECOND SUBSYNCHRONOUS BENCHMARK DATA
o
 n
 n
 o
 
o
 
o
n
118
This appendix contains the EMTP data file used to simulate 
the IEEE Second Subsynchronous Benchmark [29] . A dynamic 
brake is connected to the terminals of the generator to 
facilitate damping of shaft torsionals. The control subroutine 
described in Appendix C must be linked with the EMTP in order 
to provide control of the Type 60 sources and the 
TAGS-controlled switches shown in this data file. More 
information on the control interface can be found in Chapter 
2 and Appendix C . Figure 45 shows the benchmark in EMTP data 
format.
c ---------------------------------------------------------------------------------------
C SubSynchronous Resonance Test Case as set up by Matt Donnelly Aug 1990 I
C This is a 4-mass SSR case I
C The data comes from a case reported in the paper: I
C I
C IEEE Subsynchronous Resonance Task Force of the Dynamic System Performance I 
C Working Group, Power System Engineering Committee, "Second Benchmark Model I 
C for Computer Simulation of Subsynchronous Resonance,* IEEE Transactions on I 
C Power Apparatus and Systems, Vol. PAS-104, May 1985. I
C I
C The SSR event is triggered by a fault (simulated by a breaker closing) and I 
C the subsequent opening of a breaker clearing the fault. I
C This leads to an interaction between the resonant frequencies of the shaft I 
C and the subsynchronous resonant frequencies of the network which exist I
C due to the presence of series capacitors. I
C I
C Turbine shaft damping factors or as specified in the paper under the I
C heading "Torque Amplification Studies". I
C I
C A resistive dynamic brake is modeled to study damping of the torsinal I
C modes. The brake is switched by thyristors which are TACS controlled. I
C I
C The TACS thyristor control signals are meant to be controlled from an I
C outside subroutine which is-called from the EMTP module CSDPB. I
C I
C --------------------------------------------------------------------------- v------------
C
BEGIN NEW DATA CASE
.............................. Miscelaneous data ........................... . ....
DeltaT<-- TMax<---XOpt<---COpt<-Epsiln<-ToIMatc-TStart
.0001 2.0 60. 60.
— I0ut<— IPlot<-IDoubl<-KSSOut<-MaxOut<-- IPun<-MemSave ICate-NEnerge-IPrSup
1000 25 I I I 0 I
— KChge-- Multe---KChge---Multe--- KChge---Mul te---KChge Mul te---KChge---Mult
5 5 20 20 100 100 500 500 2000 2000
Figure 45 EMTP Data File for IEEE Benchmark
119
TAGS HYBRID 
C S_block 
C Sources ■
C ---------------------------
C The control subroutine calls for voltage at the generator terminals 
C and generator velocity. Voltages are picked up in array XTCS which 
C carries TAGS supplemental variables. Generator velocity is picked up 
C in array ETACS. The following lines place the voltages into TAGS 
C supplemental variables.
C --------------------- ------
C 345678-1-2345678-2-234S678-3-2345678-4-2345678-5-2345678-6-2345678-7-2 
9 OGENERA 60.0 -1.00 9999.
90GENERB 60.0 -1.00 9999.
90GENERC 60.0 -1.00 9999.
92GENVEL
C Supplemental Variables and Devices
9 8 TEMPVA = GENERA
98TEMPVB = GENERB
98TEMPVC = GENERC
9 8VSRCEA = 0 . 0
C 9 8VSRCEA = GENERA
C -------------------------------------------
C The control subroutine passes back the values for the Type 60 
C sources which are used in the brake legs. These are passed back into 
C array XTCS which contains TAGS supplemental variables. The following 
C lines initialize the correct XTCS entries.
C --------------------------------------------
98VSRCEB = 0 . 0  
98VSRCEC = 0 . 0  
98BRKOPN = 0 . 0  
C Outputs 
33VSRCEA
C Initial Conditions 
BLANK end of TAGS 
C
C .............................. Circuit data ..................................
C 3-phase coupled RL branch.
C Busl->Bus2->Bus3->Bus4--><----R<— ————————L
5IBUS OABUS IA 3.50 75.00
52BUS OBBUS IB 3.50 75.00
53BUS_0CBUS_1C 
5IBUS IABUS 2A 46.50 525.00
52BUS IBBUS 2B 16.75 184.75
53BUS_1CBUS_2C 
5IBUS CABUS 2A 55.00 600.00
52BUS CBBUS 2B 18.50 200.00
53BUS_CCBUS_2C 
C Series RLC branch.
C Busl->Bus2->Bus3->Bus4-- X ---;R< — — L<--- C
BUS_1ABUS_CA 9091.
BUS IBBUS CB 9091.
BUS_1CBUS_CC 9091.
GENERA_NGH_A 0.01
GENERB_NGH B 0.01
GENERC_NGH_C 0.01
C
C Jumpers for thyristor connections 
_NGH_ATMP_A1 0.01
TMP_A2 BRAKEA_NGH_ATMP_A1 
_NGH_BTMP_B 1_NGH_ATMP_A1 
TMP_B2 BRAKEB_NGH_ATMP_A1 
_NGH_CTMP_C1_NGH_ATMP_A1 
TMP_C2 BRAKEC_NGH_ATMP_A1 
C
Figure 45 Continued
120
================ BEGIN DYNAMIC BRAKE =========================
345678-1-2345678-2-2345678-3-2345678-4-2345678-5-2345678-6-2345678-7-2 
probing = 4 percent
BRAKEAVSRCEA 20.000 I
working = 16 percent is 5 ohms
a disturbance caused by a 5 ohm brake is about 16% of the gen rated MVA 
BRAKEAVSRCEA 3.000 I
BRAKEBVSRCEBBRAKEAVSRCEA 
BRAKECVSRCECBRAKEAVSRCEA 
C 
C
C Saturable transformer components.
C ------------> B u s 3 - x ---- x ---- I<--PhiBusStx-Rmag<------------------------------ >0
TRANSFORMER TRAN A
C ------- current<------------ flux
9999
C Busl->Bus2 —X ---------------------------- X --------Rk< --------Lk< --------Nk< ----------------------------------------------------------------------------------------------->0
IGENERAGENERC 0.001 0.1 22.
2BUS_2A 0.5 50.0 288.68
C < -----------> B u s 3 - x ----------------- >BusSt>
TRANSFORMER TRAN A TRAN B
C Busl->Bus2->
IGENERBGENERA
2BUS_2B
C < -----------> B u s 3 - x -------------
TRANSFORMER TRAN A 
C Busl->Bus2->
IGENERCGENERB 
2BUS_2C 
C
C Series RLC branch.
C Busl->Bus2->Bus3->Bus4-x--- R<
BLANK End of circuit data 
C
C .............................. Switch data ....,................... . ..............
C Bus— >Bus— x  Tclose<----Topen<------- Ie O
BUS_2A .01661667 .09161667
BUS_2B .01661667 .09161667
BUS_2C .01661667 .09161667
BUS_0AINFINA -1.0 9999. I
BUS_0BINFINB -1.0 9999.
BUS_0CINFINC -1.0 9999.
C ======================= TACS CONTROLLED SWITCHES FOR BRAKE ===============
C Type 11 Switches are TACS Controlled Switches. These are SCR's.
C 345678-1-2345678-2-2345678-3-2345678-4-2345678-5-2345678-6-2345678-7-2 
11TMP_A1BRAKEA BRKOPN I
11TMP_A2_NGH_A BRKOPN
11TMP_B1BRAKEB BRKOPN
I1TMP_B2_NGH_B BRKOPN
I1TMP_C1BRAKEC BRKOPN
I1TMP_C2_NGH_C BRKOPN
C
BLANK End of switch data 
C
C .............................. Source data ........................................
C Bus— xI<Amplitude<Frequency<--TO I Phi0<---O=PhiO <----Tstart<---- Tstop
>BusSt> 
TRAN C
-L<----C
14INFINA 389997. 60. -40.81293 -I.
I4INFINB 389997. 60. -160.81293 -I.
14INFINC 389997. 60. 79.18707 -I.
6 OVSRCEA I -I. 9999
6OVSRCEB I -I. 9999
60VSRCEC I -I. 9999
Figure 45. Continued
121
c
-c
C Dynamic synchronous machine.
C Bus— > < ----Volt<----- Freq<----Angle
59GENERA 17962.9 60. 0.0
C Bus— > < ----Volt<---------x ----Angle
GENERB
GENERC
C
C Machine parameter cards.
C --------- x --EPSut>A<---EPOmeg<-- EPDgEl<---------x ---NIOMax
TOLERANCES 20
C ----------------------- x ----- FM
■PARAMETER FITTING I.
C'
C Electrical parameters of machine.
C <-<-<-NP<----SM0utP<---SMOutQ<---- RMVA<------ RKV<------------ AGLine< Sl< -S2
4 2 1 2  I. I. 600.0 22. -1800. 1907. 3050.
C Col: (1-2) NuMas, (3-4) KMac, (5-6) KExc.
C ------- x ------ ADlc------ AD2<------ AQ 1<------ AQ2 <----- AGLQe------ SlQe------ S2Q
-I.
C If S.M. is not saturable (AGLine >= 0), leave SI - S2Q blank.
C
C Manufacturer supplied p.u. data.
C ------ Rae----- - -Xle------- Xde------- Xge------ X ' de------ X ' qe------ X" de-------X"q
.0045 .14 1.65 1.59 .250 .460 .20000 .20000
C T' dOe T' qOe------T B dOe----- T " qOe------- XOe--------Rne------- Xne------- Xe
4.5 .55 .040 .09 .13
C
C Mass cards
C < — --- x ---- ExTrse ----- HICOe- -----DSR-C--- ---DSMe------ HSPe
I .001383 34.4 4.39000
2 .176204 4383.2 97.9700
3 .40 .310729 7729.6 50.12823
4
C col:
.60
: (1-2) ML
.049912 1241.6
BLANK card teminates mass card.
C
C Machine output request.
C @@e— X --Nle--- N2e---N3e---N4e---N5e---N6e---N7e---N8e N9e— NlOe— Nile— N12
31
41
C col: (3) Group, (4) All
BLANK card terminates output request.
C TAGS input cards.
C ------------------------------------------
C The following card enters the gen velocity into array ETACS to be picked up 
C by the control subroutine.
C ------------------------------------------
C Bus— x --- x K I
7 4GENVEL 6
C col: (1-2) KK - either 71, 72 , 73 or 74 
FINISH
BLANK End of source data
C
C ..............;..............  Output Request ......................................
C Bus— >Bus— >Bus-->Bus-->Bus— >Bus— >Bus— >Bus-->Bus-->Bus— >Bus-->Bus— >Bus— > 
GENERA
BLANK End of output requests’
C
C .............................. Plot request .................................. .
C @@e--e-Tse-TeeYmieYMaxBus— >Bus— >Bus-->Bus— >Header--------- >Y axis---------- >
C col: (3) Graph type, (4) Units on horizontal scale, (5-7) Units per inch,
BLANK End of Plot Request 
BLANK End of All Cases
Figure 45. Continued
122
APPENDIX B
SYSTEM DATA FOR MULTI-MACHINE SIMULATIONS
123
This appendix contains the EMTP data file used to simulate 
the multi-machine system described in Chapter 2. The EMTP 
node names correspond to the bus names in the one-line diagram 
of Figure 2. The brake is implemented at the terminals of 
the Colstrip #3 conglomerate machine. The generator velocity 
is transmitted to the array ETACS through the variable GENVEL. 
The EMTP must be linked with the control subroutine shown in 
Appendix C . This subroutine provides for the control of the 
Type 60 sources and TACS controlled switches shown in the 
data file. Figure 46 shows the data file for the multi-machine 
system.
c ----------------------------------------------------------------------------------------------
C SubSynchronous Resonance Test Case adopted for EPRI RP2473-39 Oct 1989 I
C This is a 3 machine SSR case. I
C The data comes from a Montana Power Company EMTP data file supplied I
C by Ray Brush, Project Industry Advisor at MPC. I
C ’ I
C I
C The SSR event is triggered by a fault (simulated by a breaker closing) and I 
C the subsequent opening of line clearing the fault. The line is then I
C reclosed into the original fault which extinuishes itself with not further I 
C circuit breaker operation. I
C This leads to an interaction between the resonant frequencies of the shaft I 
C and the subsynchronous resonant frequencies of the network which exist I
C due to the presence of series capacitors. I
C |
C A resistive dynamic brake is modeled to study damping of the torsinal I
C modes. The brake is, switched by thyristors which are TACS controlled. I
C I
C The TAGS thyristor control signals are meant to be controlled from an I
C outside subroutine which is called from the EMTP module CSUPB. I
C I
C ---------------------------------------------------------------------------------------
C
BEGIN NEW DATA CASE 
.000100 3.00 60. 60.
1000 50 I I I 0 I
C
TAGS HYBRID 
C S_block 
C Sources
C 345678-1-2345678-2-2345678-3-2345678-4-2345678-5-2345678-6-2345678-7-2 
C -------------------------------------------
C The control subroutine calls for voltage at the generator terminals 
C and generator velocity. Voltages are picked up in array XTCS which 
C carries TAGS supplemental variables. Generator velocity is picked up 
C in array ETACS. The following lines place the voltages into TAGS 
C supplemental variables.
92GENVEL
C Supplemental Variables and Devices
98TEMPVA = CS261A
98TEMPVB = CS261B
98TEMPVC = CS261C
98VSRCEA = 0.0
C 98VSRCEA = CS261A
C
Figure 46. EMTP Data File for Multi-Machine System.
c
90CS261A
90CS261B
90CS261C
60.0
60.0
60.0
-1.00 9999. 
-1.00 9999. 
-1.00 9999.
124
c ----------------------------------------------
C The control subroutine passes back the values for the Type 60 
C sources which are used in the brake legs. These are passed back into 
C array XTCS which contains TAGS supplemental variables. The following 
C lines initialize the correct XTCS entries.
C --------------------------------------------
98VSRCEB = 0 . 0  
98VSRCEC = 0 . 0  
98BRKOPN = 0 . 0  
C Outputs 
33VSRCEA
C Initial Conditions 
BLANK end of TAGS
< 3 1 2 3 4 5 6 7 8  
C 3456789012345.67890123456789012345678901234567890123456789012345678901234567890 
C ============== = =========TRANSFORMER DATA================= = =================
C 1 2 3 4 5 6 7 8
C --------------------------------------------------------------------- -- -------------------------------------------
C This block contains transformer data. A trasnformer is employed at the 
C generator terminals and at every 230 kv to 500 kv interface bus.
C ---------------------------------------------
C 345678901234567890123456789012345678901234567890123456789012345678901234567890
51CS500A .40196 136281.867
52CS230A .10039 59831.1654 .11677 26274.2463
53CS341ACS341B -.0685 14801.5902 -.0398 6499 .64371 .01729 1608.93290
51CS500B .40196 136281.867
52CS230B .10039 59831.1654 .11677 26274.2463
53CS341BCS341C -.0685 14801.5902 -.0398 6499 .64371 .01729 1608.93290
51CS500C .40196 136281.867
52CS230C .10039 59831.1654 .11677 26274.2463
53CS341CCS341A -.0685 14801.5902 -.0398 6499 .64371 .01729 1608.93290
51BV500A .30543 131311.152
52BV230A .06864 57649.8716 .08938 25316.0989
5 3 BV3 41ABV3 4IB -.0502 14262.1651 -.0296 6262 .72945 .01277 1550.237553
51BV500B .30543 131311.152
52BV230B .06864 57649.8716 .08938 25316.0989
53BV341BBV341C -.0502 14262.1651 -.0296 6262 .72945 .01277 1550.237553
51BV500C .30543 131311.152
52BV230C . 06864 57649.8716 .08938 25316.0989
5 3 BV3 41CBV3 4IA -.0502 14262.1651 -.0296 6262 .72945 .01277 1550.237553
51CS230A .03101 11398.5604
52CS221ACS221B -.0049 1791.69765 .00077 281. 771773
51CS23 OB .03101 11398.5604
52CS221BCS221C -.0049 1791.69765 .00077 281. 771773
51CS230C .03101 11398.5604
52CS221CCS221A -.0049 1791.69765 .00077 281. 771773
51CS230A .03101 11398.5604
52CS222ACS222B -.0049 1791.69765 .00077 281. 771773
51CS230B .03101 11398.5604
52CS222BCS222C -.0049 1791.69765 .00077 281. 771773
51CS230C .03101 11398.5604
52CS222CCS222A -.0049 1791.69765 .00077 281. 771773
51CS500A .08423 41874.9137
52CS261ACS261B -.0071 3507.66603 .00059 293. 943793
51CS500B .08423 41874.9137
52CS261BCS261C -.0071 3507.66603 .00059 293 .943793
51CS500C .08423 41874.9137
52CS261CCS261A -.0071 3507.66603 .00059 293. 943793
51GA500A .08399 46050.4691
52GA230A -.0368 20170.8879 .01613 8841 .80996
51GA500B .08399 46050.4691
52GA230B -.0368 20170.8879 .01613 8841 .80996
51GA500C .08399 46050.4691
52GA230C -.0368 20170.8879 .01613 8841 .80996
51HS500A .08399 46050.4691
52HS230A -.0368 20170.8879 .01613 8841 .80996
51HS500B .08399 46050.4691
52HS230B -.0368 20170.8879 .01613 8841 .80996
51HS500C .08399 46050.4691
52HS230C -.0368 20170.8879 .01613 8841 .80996
Figure 46. Continued
O
O
O
O
125
.========================SERIES CAPACITOR DATA=============================
1 2  3 4 5 6 I  8
345678901234567890123456789012345678901234567890123456789012345678901234567890
BCSWlACB3 41A 50916.
BCSW1BCB3 4IB 50916.
BCSW1CCB341C 50916.
BCSW2ACB342A 50916.
BCSW2 BCB3 4 2 B 50916.
BCSW2 CCB3 4 2 C 50916.
GBSW1ABG75IA 24887.
GBSW1BBG7 5IB 24887.
GBSW1CBG7 5IC 24887.
GBSW2ABG752A 24887.
GBSW2BBG752B 24887.
GBSW2 CBG7 5 2 C 24887.
GTSWlAGTO OlA 34680.
GTSWlBGTOOlB 34680.
GTSWlCGTOOlC 34680.
GTSW2AGT002A 34680.
GTSW2 BGTO 0 2 B 34680.
GTSW2 CGTO 0 2 C 34680.
C 1 2 3 4 5 6 7 8  
C 345678901234567890123456789012345678901234567890123456789012345678901234567890 
CBOOlACBFFlA 0.01
C CBO OlBCBFFlBCBO 0IACBFFlA 
C CBOOICCBFFlCCBO 0IACBFFlA 
C CB0 0 2 ACBFF2 ACB 0 01ACBFFIA 
C CB 0 0 2 BCBFF2 BCBOOIACBFFlA 
C CBO 02CCBFF2CCB001ACBFF1A 
C CB3 41ABCFF1ACB00 IACBFFlA 
C CB3 41BBCFF1BCB00IACBFFlA 
C CB3 41CBCFF1CCB0 0IACBFFlA 
C CB3 4 2 ABCFF2 ACBOOIACBFFIA 
C CB3 4 2 BBCFF2 BCBOOIACBFFIA
C CB342CBCFF2CCB0OlACBFFlA i
BGOOIABGFF1ACB00IACBFFlA 
BGO OlBBGFFlBCBOO IACBFFlA 
BGOOlCBGFFlCCBOOIACBFFlA 
C BG002ABGFF2ACB001ACBFF1A 
C BG 002 BBGFF2 BCBOOIACBFFlA 
C BG 002 CBGFF2 CCBOOIACBFFIA 
C BG751AGBFF1ACB0OlACBFFlA 
C BG751BGBFF1BCB0 01ACBFF1A 
C BG7 5ICGBFFlCCBO 0IACBFFlA 
C BG7 52AGBFF2ACBO 01ACBFF1A 
C BG7 5 2 BGBFF2 BCBOOIACBFFlA 
C BG7 52CGBFF2CCB0 01ACBFFlA 
C GTO 0IAGTFFlACBO 0IACBFFlA 
C GTO 01BGTFF1BCB00IACBFFlA 
C GTO 01CGTFF1CCB00IACBFFlA 
C GTO 02AGTFF2ACBO 0IACBFFlA 
C GTO 0 2 BGTFF2 BCBOOIACBFFlA 
C GT002CGTFF2CCB001ACBFF1A 
C GT3 91ATGFF1ACB00IACBFFlA 
C GT3 9IBTGFFlBCBO 0IACBFFlA 
C GT3 91CTGFF1CCB00IACBFFlA 
C GT392ATGFF2ACBOOlACBFFlA 
C GT392BTGFF2BCB001ACBFF1A 
C GT3 9 2 CTGFF2 CCBO 01ACBFFlA
Figure ,46. Continued
w
w
w
w
w
w
w
w
w
w
w
w
w
w
w
w
w
w
n
n
 
o
 
n
 
o
o
o
o
 
o
n
o
126
C INSERT JUMPERS HERE AS NEEDED======
C these jumpers are for the brake 
CS2 61A_NGH_A 0.01
CS2 61B_NGH_BCS2 61A_NGH_A 
' CS2 61C_NGH_CCS2 61A_NGH_A 
C Jumpers for thyristor connections 
_NGH_ATMP_A1 0.01
TMP_A2 BRAKEA_NGH_ATMP_A1 
_NGH_BTMP_B1_NGH_ATMP_A1 
TMP_B2 BRAKEB_NGH_ATMP_A1 
_NGH_CTMP_C1_NGH_ATMP_A1 
TMP_C2BRAKEC_NGH_ATMP_A1
C ================ BEGIN DYNAMIC BRAKE =========================
C 345678-1-2345678-2-2345678-3-2345678-4-2345678-5-2345678-6-2345678-7-2 
C probing = 4 percent
BRAKEAVSRCEA 10.000 I
C a disturbance caused by a 5 ohm brake is about 9% of the large gen MVA
C BRAKEAVSRCEA 5.000 I
C BRAKEAVSRCEA 3.500 I
C BRAKEAVSRCEA 1.700 I
C BRAKEAVSRCEA 0.850 I
C BRAKEAVSRCEA 0.425 I
C BRAKEAVSRCEA 0.212 I
BRAKEBVSRCEBBRAKEAVSRCEA
BRAKECVSRCECBRAKEAVSRCEA
========================STRAY CAPACITANCE DATA============================
1 2 3 4 5 6 7 8  
345678901234567890123456789012345678901234567890123456789012345678901234567890 
CS221A 37.699
CS221B 37.699
CS221C 37.699
CS222A 37.699
CS222B 37.699
CS222C 37.699
CS341A 37.699
CS341B 37.699
CS341C 37.699
BV341A 37.699
BV341B 37.699
BV341C 37.699
1 2 3 4 5 6 7 8  
3456789012345678.90123456789012345678901234567890123456789012345678901234567890 
AUX-LOAD=================AUX-LOAD DATA============================
AUX-LOAD CSl
13.5535.8729
13.5535.8729
13.5535.8729
CS221A
CS221B
CS221C
AUX-LOAD CS2
CS222A
CS222B
CS222C.
AUX-LOAD CS3&4 
CS261A 
CS261B 
CS261C
13.5535.8729
13.5535.8729
13.5535.8729
4.2118.94766
4.2118.94766
4.2118.94766
=END OF AUX-LOAD DATA=
Figure 46. Continued.
127
LINE-MODEL 
ENGLISH
C 
C 
C 
C 
C 
C 
C 
C 
C 
C 
C 
C 
C
C LINE LENGTH 
C 
C
-1CB001ACB341A
===BEGIN LINE DATA= 
FD-LINE
0
0
0
0
1
2
3
4
5
6
.5
.5
.5
.5
.5
.5
.5
.5
.5
.5
1.217
1.217
1.217
1.217 
.111 
.111 
.111 
.111 
.111 
.111 
= I.
4
4
4
4
4
4
4
4
4
4
8363E+02
.0355
.0355
.0355
.0355
1.140
1.140
1.140
1.140
1.140
1.140
-90.25
-49.75
53.00
87.00 
-92.00 
-70.00 
-48.00
51.00
70.00
89.00
102.
102.
97.
97.
58.
88.
58.
58.
86.
58.
102.
102.
97.
97.
58.
88.
58.
58.
86.
58.
KM
18.
18.
18.
18.
18.
18.
TRANSFORMATION MATRIX AT F 6.0000E+01 HZ
4.35906631896525723846E+02
-2 6
2.338503266862541992D+02 
7.576834382353307092D+03 
1.553934907365646995D+04 
1.303277879477250099D+04 
1.051367575432140584D+06 
1.788800464233595878D+07 
3.215552750799266935D-03 
I .502368897665527586D+00 
8.883134776932094567D+01 
3.231483173792362109D+02 
4.373782152485587267D+04 
I.720826673075218219D+06
4.607025887907407764D+03 -I.120704261724998764D+04
2.371811402242927302D+00 ' -----" ------- -----
1.239955896651661647D+03 
6.041075105998212530D+04 
3.173782052948331169D+06 
9.697832386580994166D+07 
1.366422002065763447D+00 
I .739261577341605358D+00 
9.894953912820452757D+01 
2.360165964307623142D+03 
I.391197319580966760D+OS 
9.501318499961066293D+06
I.626077136742044360D+03 
1.855115543686415003D+04 
3.565511897423155751D+05 
9 .772944967024798039D+06
I .481937981804738791D+00 
I .813361892790934515D+01 
I .699077331332729806D+02 
1.401695415313843205D+04 
4.543061779895553482D+05
6.71020350886229139460E-04
2.011900277981229317D-05 
4.58889886008457203ID-03 
5.079103839811955623D-01 
3 .344846128277546848D+00 
4.565945987609417287D+02 
3.133956389415313197D+03 
-7.007366106724821962D+06 
I.985910919608001356D-02 
6.893476457704126426D+00 
8.554248425508355247D+01 
1 .927858339948064241D+02 
5.355560189399535489D+03 
I.313151681067512368D+04 
3.190298029200530436D+04 
-2CB001BCB341B
5.251917054175893524D-04 
3.673213967943864382D-02 
2.2731227215435853OlD+00 
1.787365926589398724D+01 
9.838432007534183725D+02 
8.071949154778653065D+04 
5 .403756702878579199D+04 
I.579274337661374733D+00 
2.71338481884075282ID+01 
I.059222595596668022 D+02 
8.203738463800586800D+02 
6.919590712767171567D+03 
2.573539378358522890D+04 
4.320249960627566361D+04 
I
6.14318893868977462OD-04 
I.310920694691021342D-01 
2.322591614930783699D+00 
1.793874642721145953D+02 
2.605599258416033251D+03 
6.865223171533790417D+06
I .854493286967705545D+00 
4.766789109899327759D+01 
1.791866081532796144D+02 
3.048684192543276993D+03 
I .067565497313505OllD+04 
3.187140841388348736D+04
3.6191270537147129715SE+02
2.179095507925209141D+02 
-3.398942756219749484D+02 
4.3113-689377714563020+02 
1 .169167130888102086D+02 
3.076163302584957250D+03 
7.731944862554415129D+06 
3.OS8240977417195513D-03 
I .985593179875473319D+00 
8.075819525738295424D+00 
I.742496285891184282D+01 
5.066481626583442832D+02 
I .672657578545732918D+06
1.917622208210564708D+03 
1.953576204285552720D+02 
3 .746789506457803896D+02 
2.289318258532243426D+02 
I .148294637484817904D+05 
4.253656963687165827D+07 
I.353025611085977992D+00 
2.087500541855408176D+00 
9.881227064003834215D+00 
2.914809137316892373D+01 
2.4120611181328542440+04 
9.311039332295559580D+06
-2 6
-5.491042173376289242D+02 
4.782172719259605298D+01 
-9.699771043113183566D+01 
4.178014062019370499D+03 
3.1526671991589OS656D+06
I.885315614318739091D+00 
2.965369900935183600D+00 
I.040632849003995930D+01 
4.919127056615716143D+02 
3.334005532842211396D+05
6.27220272007161967764E-04
2 .428401412499489389D-05 
I .605775757898926574D-03 
2 .575111574861040347D-02 
1.984514141022057226D-01 
1.756463180569881843D+03 
I.758079464863565772D+OS 
•2.4859294762691185950+10 
1.985702309233852454D-02 
2.8122892576376982140+00 
8.943993577573018605D+01 
2.115611873451258376D+02 
2.485146854825878563D+04 
1 .02 2 54 9 04 3 3 03 5153 9 ID+05 
1.753143284423927362D+05 
-3CB001CCB341C
I -215739715150534003D-03 
3.091474673101401256D-03 
8.306539628916730263D-02 
I.140396655863642383D+01 
7.914873814780635939D+03
1.248684470309135779D-03 
1.558653377089461774D-02 
9.918909329150904230D-01 
2.968556269168396042D+02 
5.261418445948879253D+04
2.475768142174146175D+10 -4.9617211684538328170+10
-3.129715794945099551D+05 
2.091957570410926837D+00 
5.412674814600893058D+00 
1 .437732943629891551D+02 
1 .220525766209303868D+03 
4.080934151466981621D+04 
1 .7496751044727694900+05 
1.020979665995474104D+06 
I. -2
3 .100686356069762187D+05 
2.19659245535777553ID+00 
5.539074440068783645D+01 
2.0648263027750205010+02 
I.187063330236078514D+04 
6.972219484653677682D+04 
I.751408335977126917D+05 
I.021991050405007307D+06 
6
Figure 46. Continued.
^
128
2.36466745833385267872E+02
2.027334571071493201D+02 
-4.2618282121959381040+01 
3.9870165123023394930+02 
-1.1186599930723410240+03 
1.2410391526999090140+02 
6.3729661494713957650+05 
2.7512878122188370480-03 
1.7628204850284546200+00 
5.585368779717997167O+00 
1.191691385218775134O+01 
4.3226070151761574860+01 
7.0530812336178707480+05
20 6.12843377942087064311E
8.1887758399558642000+02 
3.7378827080820999920+01 
2.9421406821500294630+02 
9.6489926114922958790+01 
I.101964541538011328O+02
I.149531578482481858O+00 
2.200321601762747492O+00 
8.8486476675370526750+00 
1.6356094447112032860+01 
7.0736775055061405840+01
2.3398910661327510140+02 
-1.8677938602473690140+00 
1.4764089280921314470+03 
I.3828937750406822720+02 
5.7483160319041854790+03
1.4738235608100805350+00
2.6678049728834724030+00
1.1721499878815831770+01
2.7356701210052876140+01
6.346552264856925603D+03
3.0805124025238135960-05
5.9282154847607025250-03
7.6510071031221655710-03
4.4690766004208473560+01
6.2708824131359502870+03
1.7758331907485974370+06
2.9911718864958648090+09
1.9853755778278079840-02
4.3214011869173706290+00
4.5460885363974061770+01
4.0641783913076694150+03
1.2818399304697324260+05
1.6711274110592964570+06
3.7657365504387110120+06
-4CB002ACB342A
1.1878142748987249940-03 
I .1337163350168419150-02 
5.007326771328374140O-02 
7.7641514172253817350+01 
8.1361365166249222970+04 
5.1645386090075900550+09 
-2.9426209201008423570+09 
1.7182376455760820510+00 
8.2718207139283979060+00 
2.9679745706925869090+02 
1.3941470695339168510+04 
5.0558332935162421200+05 
3.1774469773974852290+06 
3.7694668962667146120+06 
I.
1.3191517863103259750-03 
I .0328223505209593000-02 
1.6378977300667177040-01 
8.1243326379377823800+02 
3.1436507436153374870+05 
-5.2152683409325008390+09
1.9303973364732445740+00 
1.5147633026139227350+01 
4.753434228112728022O+02 
3.5981931106316821570+04 
8.8203782164855304290+05 
3.1805945624494606160+06
2.38272873525244705206E+02
2.0120290851393687600+02 
-2.9718385907313655810+01 
2.8684407134635743830+02 
1.0966647293030387540+02 
5.9651652394486569620+03 
2.7609687184670015880-03 
I.758765335951147174O+00 
5.0274536726647011120+00 
1.6590225516737128860+01 
3.3409548035952826690+03
8.2241268520814072930+02 
2.9803613807913504450+01 
3.5227357947243005470+03 
1.412312295716831336O+02 
2.6802779639813265530+06 
1.1489620380953853850+00 
2.183241811585384495O+00 
1.0406302659188365170+01 
2.755372978813364426O+01 
1.5063145829442493850+06
-2 6
2.1242652742719900160+02 
1.6445623412409040580+01 
-2.8113976893783063820+03 
1.2769203092636195860+02
1.4650279275503435010+00 
2.7082515286174561050+00 
1.0572306841379637500+Oi 
4.5112242223350091490+01
6.13639970723752964442E-04
3.0461239378479168400-05
4.6529448868376211380-03
4.8776337968025888950-03
9.7890171856546865920+00
1.2480775756049813940+04
1.7699841562681223440+06
1.1309522207796684290-03 
1.0269120429719091330-02 
1.0304919579367441670-02 
8.7358611572937592270+01 
2.3523430642409484340+04 
I .6820640135545016450+08
1.6607124113101751730+08 -1.6437024922170160340+08
1.1506343539367375180-03
7.6724625345095258430-03
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Figure 46. Continued.
129
3.138346725035655755D-05 
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4.2420864212807234140+06 
5.293649250039050635D+07 
= 0 )
0.51440260 -0.40757985 
0.00000000 0.00000000 
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- 0 .
0.
0.
0.
-0.
0.
C LINE-MODEL FD-LINE
C ENGLISH
C 0 .5 1.217 4 .0355 -90
C 0 .5 1.217 4 .0355 -49
C 0 .5 1.217 4 .0355 53
C 0 .5 1.217 4 .0355 87
C I .5 .111 4 1.140 -92
C 2 .5 .111 4 1.140 -70
C 3 .5 .111 4 1.140 -48
C 4 .5 .111 4 1.140 51
C 5 .5 .111 4 1.140 70
C 6 .5 .111 4 1.140 89
C LINE LENGTH = 2.. 1469E+02 KM
25 102. 102.
75 102. 102.
00 97. 97.
00 97. 97.
00 58. 58. 18. 4
00 88. 88. 18. 4
00 58. 58. 18. 4
00 58. 58. 18. 4
00 86. 86. 18. 4
00 58. 58. 18. 4
Figure 46. Continued.
O 
O
13 0
TRANSFORMATION MATRIX AT F 6.0000E+01 HZ
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Figure 46. Continued
131
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3.2437909900705311880+05 
1.1609053062462728280+08 
-3.7584154957439611110+08 
1.9850809710734844750-02 
6.4502242725718365040+00 
4.464182829413816194O+03 
8.7110208578335914350+04 
3.7379306191958992710+05 
1.1748674138494223590+06 
4.0268234531641748620+06 
5.8058723587378812950+06 
-6BG002CTN152C
2.1421990459158251910-03 
9.4549935760919903790-03 
4.3668369061717942260+02 
8.4576987632914265300+03 
5.6706026312310452340+04 
3.8456873286378339250+05 
-1.1567825440144047700+08 
8.0302459029097057880+07 
1.6109052095361392210+00 
7.2419804192085859280+00 
2.0081746498781323230+04 
1.5934037461244637960+05 
5.0820529137005188390+05 
1.5868061877914610090+06 
4.0308124321772893890+06 
8.8319325265812017020+06 
I.
4.882162107648864327O-03 
2.570312526798858466O+00 
1.3489570006716944310+03 
1.3389784533697735010+04 
1.2721608781618026110+05 
-1.1150042229655062690+05 
3.7334753728379835190+08 
-7.9076002796477966010+07 
3.6709935549728082150+00 
4.8620558825233864300+02 
5.7727204215151922650+04 
2.4096903913607484360+05 
7.0415071764369808080+05 
2.1171081321827335050+06 
5.8001267420957003490+06 
8.8406814558311533180+06 
-2 6
Figure 46. Continued.
132
i
i
4
1
3
2
I
5 
I. 
I.
3.
6.
4. 
I. 
I. 
8.
1.
2.
-3 .
I.
5.
4.
6. 
I.
5. 
I. 
3.
6.
: 0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
2.22691865734314387026E+02
. 985264660626852269D+02 
.9597780939891657150+01 
.2073962029571487160+02 
.1885109343378432630+02 
.3091016852383985450+03 
.7152568517646785220-03 
.7745843668777943890+00 
.8942898992519304760+00 
.7788025741618579630+01 
.9348177585664928980+03
9.1118702112936648520+02 
3.2373473781062339860+01 
8.3697415472661654690+02 
1.5269525203876804300+02 
2.4174545636141609790+05 
I.1733836925299033770+00 
2.2131054192928757970+00 
I.0383139009225360590+01 
3.0915809761604156060+01 
1.4160811471926270680+05
4.6482627899483800250+01
1.0302941089031447670+00
-2.139527119861914635O+02
3.1060480018001523210+02
1.4441902558231844710+00
2.7004341612083599400+00
1.1634445684409164600+01
9.8529657167572899820+01
7.16277452086275655155E-04
6931993644332597670-05 
8833678606083586020-03 
6145104452044985340+01 
7919228333637706780+03 
9795170006535736770+04 
4987240270418273210+04 
2488444418370522180+05 
1277658197280157730+08 
0151116184193241410+07 
9850684386600679370-02 
2494026158953419610+00 
3355647604583012940+03 
4850847356769325420+04 
9224492969120656930+05 
3440597665790893370+05 
1031539149866796800+06 
5592912792757780410+06 
2591912789662868020+06 
Q MATRIX BY ROWS (IMAGINARY PART 
-0.54250481 -0.31965924 
0.00000000 
-0.32366413 
0.00000000 
-0.29410167 
0.00000000 
0.29655495 
0.00000000 
0.35902679 
0.00000000 
0.51378486 
0.00000000
2 .555176672514717403D-03 
1.125932412704136226D-02 
3.615682032339228940D+02 
3.8420375950807031130+03 
4.996683969280478414D+04 
2.8718496824799328170+05 
5.344440673888587480D+05 
-2.139409561514032334D+08
1.935041128740468908D+00 
8.614286983260198260D+00 
1.638203620847675325D+04 
8.294533135325085095D+04 
3.123220773954449978O+05 
6.8199597096756409160+05 
1.644473792068105366D+06 
3.5628171200233867860+06
4.191067570570848415D-03
2.238554782500394691O+00
8.618274849980417969D+02
8.075948321913981772D+03
1.7544533476981112470+04
-1.616037198778482089D+05
-2.515439390042628911D+05
3.059484904538657237O+07
3.184183324247265023D+00
4.267991693019570647D+02
3.734087547484763309D+04
1.3822500345848764850+05
3.592068224273715314D+05
7.3272233663128336780+05
2.127509250437912415D+06
6.252997048132930300D+06
44364168
00000000
34003824
00000000
43251837
00000000
42289065
00000000
34832059
00000000
43388185
00000000
0.00000000
-0.03778065
0.00000000
= 0)
0.51440260
0.00000000
0.01099281
0.00000000
0.54988139 -0.53854693 
0.00000000 0.00000000
-0.64921677
0.00000000
0.05622246
0.00000000
0.40262051
0.00000000
-0.43042328
0.00000000
-0.00653088
0.00000000
0.48850481
0.00000000
-0.40757985
0.00000000
0.76017469
0.00000000
-0.24036744
0.00000000
0.07468529
0.00000000
-0.36512374
0.00000000
0.21781819
0.00000000
-0.12395691
0.00000000
0.35250691
0.00000000
-0.19732722
0.00000000
-0.31444056
0.00000000
0.75456892
0.00000000
- 0.36967658
0.00000000
LINE-MODEL
ENGLISH
FD-LINE
C 0 .5 2.400 4 .0250 -7.00 140.25 140.25
C 0 .5 2.400 4 .0250 7.00 140.25 140.25
C I .5 .04967 4 1.603 -15.00 52.25 52.25 18. 3
C 2 .5 .04967 4 1.603 -25.00 83.25 83.25 18. 3
C 3 .5 .04967 4 1.603 -15.00 114.25 114.25 18. 3
C 4 .5 .04967 4 1.603 15.00 52.25 52.25 18. 3
C 5 .5 .04967 4 1.603 25.00 83.25 83.25 18. 3
C 6 .5 .04967 4 1.603 15.00 114.25 114.25 18. 3
LINE LENGTH = 1.6270E+02
TRANSFORMATION MATRIX AT F
C 
C 
C
-1TN151ABG751A
KM
6.0000E+01
17 6.22346036394520879753E+02
-2 6
1.998551292256145118D+02 
7.960273288430282435D+02 
9.012256877433020691D+03 
5.983994606779328751D+04 
3.194143710404731624D+06 
3.159442663144744'514D+07 
3.64253 8396971592765D-03 
I .354526317672298474D+00 
I.050412119616732127D+02 
I .106916256056727690D+03 
6.I64616I83856358566D+04 
I .410570888077296288D+06
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5.716428010259349701D+01 
4.36585853993266437ID+04 
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I.880727936343301265D+00 
2 .763899081053263487D+02 
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2.452987618793558613D+OS 
8.941810527297823690D+06
-I.70870154479912972ID+03 
1.754180042507906364D+03 
1 .8 018 87 5047 4 64 9 07 3 ID+04 
4.071320657945612329D+05 
I .142854444487232040D+07
I.334029529507843592D+00 
2.085005920974407845D+01 
4.075216182611066031D+02 
I .523048702415280627D+04 
4.946415805551151279D+05
Figure 46. Continued.
133
1.049726308933693033D-05 
2.572672642788628229D-03 
2.332271855995639263D-01 
2.919793433104464708D+OO 
3.42650483514950153ID+02 
3.5124404631485720070+03 
-I.5449598636640370120+07 
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2.8932236582330310880+02 
4.6809961843037078780+03 
1.2716921098146146280+04 
3.2384626289599023490+04 
-2TN151BBG751B
6.08537864798103196118E-04
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3.6629082694934682880+04 
2.0524290584578783750+00 
1.747569183878789767O+01 
I .5503753751375850460+02 
4.7070227297344843010+02 
6.3816164738337027980+03 
2.3155408949723907880+04 
4.8549474717687056910+04 
I.
5.2938980100520911100-04 
5.751698996574868862O-02 
4.6414965482397201320+00 
8.5645793689548995080+01 
1.4625530115925284350+03 
1.5379830731304763350+07
3.3462092721635137700+00 
4.4622279804115299480+01 ' 
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I.8802440410831579810+03 
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7.1769684520450630320+03
5.44757048995063447453E-04
I.8002617440111917160-05 
9.4568934911250605300-04 
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-3TN151CBG751C
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7.4660857886184163590+02 
2.0500393615924559980+04 
1.7214960457762324590+05 
7.1005577129251747100+05 
I.
-2 6
4.2960031754212272940+01 
1.3038754413689343790+00 
-5.4850742948332354840+02 
6.370937884452725644O+06 
1.7704737274829381870+00 
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3.4959001239715756240-01 
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-2 6
2.73243042762773065135E+02
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5.43988667097261867139E-04
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-4TN152ABG752A
I.0670070046627054680-03 
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4.2007273778110518380+03 
1.1358377087987648700+05 
6.1915935339674113490+05 
1.4470448677505654630+06 
I.
-1.2188039451229226420+04 
1.9773129388126748300+02 
7.6187603663774323690+01 
6.0193381455366752340+06 
1.6475139350511790740+00 
5.2970315946850540230+00 
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3.9461761897403852780+06
1.5606210434632815220-03 
7.2286409526320559110-04 
6.5186227157771538250-02 
1.6069883266731569420+00 
1.9588750569808975400+02 
6.5354094160263627600+04 
6.420503895895764232O+08
2.6255520558374763620+00
9.7871346835032830840+00
4.4380850655372829290+02
1.3347316083027415350+03
1.9032832917641203490+04
2.3962194833646100600+05
1.0736874644899848500+06
-2 6
Figure 46. Continued.
134
2 .20177781503156975162E+02
1 .53 3 42 9 94 9195 67416 90+02 
-I.1337424806305393470+03 
6.1906238050969538730+02 
8.0631876443797688480+01 
1.2797038786930137800+04 
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9.2196935790848622220+03
23 5.42722536070877845126E-04
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6.930644630376004613O+02
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-5TN152BBG752B
I .1804082782402128570-03 
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1.2997005838852692700+04 
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4.3277172132585217240+07 
I.
I .7015119021647301880+02 
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2.083988024309602297O+05 
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5.129785539001889356O+04 
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1.3838735311964512570+07
2 .33212776567026324415E+02
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3.2565735794866019860+02 
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6.8301716057848020470+01 
2.7433555016360898190-03 
1 .8672691369632636040+00 
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1.0008691011699775800+04 
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1.5476493631996910140+00 
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1.2038205667102330750+01 
5.2118828474229172800+02
5.4273630,9301992278183E-04
2.0828538289812162730-05
3.7742284616539122780-03
2.2134034576978072000-02
1.6911705653180154340+00
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6.7078820199834641240+04
1.2664857676690562000-03
2.9732214606161718450-03
1.4389890151576945410-01
6.1317451496787850340+00
3.9844665044663648810+03
5.8984967290344039790+05
6.9451571016050507130+08 -6.9019382092577816550+08
-4.8289144666491993410+12
1.9858753928147778070-02
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2.7575410179804432430+02
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9.8966622354057014450+06
3.5976752531112096270+07
-6TN152CBG752C
4.6102351332775816650+12 
I.9198569912591015860+00 
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8.3266878083693354990+02 
2.3607138495377126900+03 
9.461489193831175362O+04 
2.5543035881427291320+06 
9.9089767913124652110+06 
3.6309061258505758830+07 
I.
-2 6
-9.5927538032942463810+03
4.1905937481175509160+02
8.0344690052234092550+01
1.3905095131100364600+04
1.5992873419360305040+00
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3.9926733590677456670+12
-3.7739997029231200560+12 
2.3808937089884764650+00 
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4.2352747711643381040+06 
3.5941149160024666230+07 
3.6345029084021490070+07 
-2 6
2.38428075051592937683E+02
I.562954926202384485O+02 
1.5104145384349836600+04 
-1.3842893753893300610+02 
7.2180769215806682480+01 
2.7492069186787277320-03 
1.6984339661205038460+00 
8.4209633936186567560+00 
3.1179644538158888700+01
I.2422949426016137070+04 
3.3815330156323950560+01 
6.6681692358281132730+01 
8.8754864041939477200+02 
1.5787265010572736620+00 
2.8551354925706502460+00 
1.2086977363596657310+01 
6.1432240119109611950+02
-2.6763768257296094820+04
5.9405735716945244460+02
7.7144830731491104810+01
1.8312560433208102040+04
1.6620983625157836620+00
7.6080824387320934440+00
1.9098149205273395430+01
1.2646103243562018860+04
Figure 46. Continued.
135
5.42753562233588230518E-04
2.085955599229596916D-05 
4.757015436042039570D-03 
8.591341266752282324D-03 
I.303574I05743703637D+00 
1 .440334418039532274D+02 
7.530692085646290070D+04 
6.350697090973523096D+06 
7.974715475144418422D+06 
I.985873839454865624D-02 
6.6416420523-923194170+00 
I.9469731110047246500+02 
9.6320189259897141200+02 
2.4882214936139959260+04 
7.1007764754860424730+05 
8.1798701950752893460+06 
2.2632432699111924510+07
Q MATRIX BY ROWS (IMAGINARY PART 
0.50087070 -0.46167319 -0.49313081 
0.00000000 0.00000000 
0.34049665 -0.02109095 
0.00000000 0.00000000 
0.35677436 0.53295208
0.00000000 0.00000000 
0.50087517 -0.46168710 
0.00000000 0.00000000 
0.34049490 -0.02108988 
0.00000000 0.00000000 
0.35677883 0.53297823
0.00000000 0.00000000
I .3389961420323537450-03 
I.8528123184996910440-03 
3.620576256203986501D-02 
3.379102363055385624D-01 
1.399486812621064786O+03
1.977702913359153357D-03
6.757763689435925984O-04
1.445271703749602038D+00
8.376346211240381123D+01
1.2285624242225641640+04
2.7608342514932002200+00 
1.533055705715490791D+01 
9.406065271133197996D+02 
7.3048972387928968150+03 
2.5009444802196431190+05 
3.3870743234340892520+06 
1.526254052225408982D+07
1.340665641285117075D+06 -1.5258988073205704270+04 
-6.8393868981203012170+08 6.766638950154676884O+08
-8.4652473937581898640+06 
1.862812870140722527D+00 
I .0366294399313177530+01 
4.080195400158612102D+02 
9.9883646256403314110+02 
6.061971691176961940D+04 
2.3597973673659735940+06 
1.5247436382616116200+07 
2.265485243022319302D+07 
= 0)
0.24515321 0.47310880
0.00000000 0.00000000 
-0.44697417 -0.25127029 
0.00000000 0.00000000 
0.48533423 
0.00000000 
-0.24513653 
0.00000000 
0.44693392 
0.00000000
0.00000000
-0.44756805
0.00000000
-0.18697741
0.00000000
0.49313768
0.00000000
0.44755453
0.00000000
0.18698441 -0.48531000 
0.00000000 0.00000000
-0.44765855 
0.00000000 
47333592 
00000000 
0.25174401 
0.00000000 
0.44741519 
0.00000000
-0
0
-0.30648209
0.00000000
0.58002629
0.00000000
-0.26281078
0.00000000
-0.30604273
0.00000000
0.57993411
0.00000000
-0.26322140
0.00000000
LINE-MODEL
ENGLISH
FD-LINE
C 0 .5 2.400 4 .0250 -7.00 140.25 140.25
C 0 .5 2.400 4 .0250 7.00 140.25 140.25
C I .5 .04967 4 1.603 -15.00 52.25 52.25
C 2 .5 .04967 4 1.603 -25.00 83.25 83.25
C 3 .5 .04967 4 1.603 -15.00 114.25 114.25
C 4 .5 .04967 4 1.603 15.00 52.25 52.25
C 5 .5 .04967 4 1.603 25.00 83.25 83.25
C 6 .5 .04967 4 1.603 15.00 114.25 114.25
C LINE LENGTH = 2.•5154E+02 KM
TRANSFORMATION MATRIX AT FC 
C
-IGTO 01AGT3 9IA
6.0000E+01 HZ
I.
6.22346036394520879753E+02
I .998551292256145118D+02 
7.960273288430282435D+02 
9.012256877433020691D+03 
5.983994606779328751D+04 
3.19414371040473I624D+06 
3 .159442663144744514D+07 
3 .642538396971592765D-03 
1.354526317672298474D+00 
I .050412119616732127D+02 
I .106916256056727690D+03 
6.164616183856358566D+04 
I .410570888077296288D+06
I .571346770229615146D+03 
5.7164280102593497OlD+01 
4.36585853993266437ID+04 
2.228219059471568289D+05 
6.115818617726868368D+06 
I .958277172619485036D+08 
I.245166501626172068D+00 
I .880727936343301265D+00 
2.763899081053263487D+02 
7.70332023198599552OD+03 
2 .452987618793558613D+05 
8.941810527297823690D+06
18.
18.
18.
18.
18.
18.
-2 6
-I.708701544799129721D+03 
I .754180042507906364D+03 
1.801887504746490731D+04 
4.071320657945612329D+05 
1.142854444487232040D+07
1.334029529507843592D+00
2.085005920974407845D+01
4.075216182611066031D+02
1.5230487024152806270+04
4.946415805551151279D+05
9.59182804229144359161E-04
1.622389006637656498D-05 
1.621592759115266077D-03 
I .4072465032140024300-01 
1.053181508240430775D+01 
6.003090979687323436D+01 
2.030909266766466345D+03 
-9.770702127622512635D+06 
1.986105941787055709D-02 
5.171507082780827713D+00 
5.436994790605101002D+01 
2.486778696887722440D+02 
1.366804845899864659D+03 
7.692744615701141811D+03 
2.061522063673208049D+04 
-2GT001BGT391B
2.161925607784703852O-04 
8.117305980332966504D-03 
5.827001161636710808D-01 
9.584318996460890361D+00 
5.2313418253491349220+02 
1.275233746602577025D+04 
1.515827811691109309D+04 
1.381740472126613495D+00 
1.286882083731184201D+01 
1.081505631362237185D+02 
3.951409553438248210D+02 
3.7392809671698708480+03 
1.414977825460381791D+04 
3.460881497732154912O+04 
I. -2 6
2.844485765432061393O-04
2.745650994227643660D-02
2.254827786129737666D+00
1.424536892776863950D+01
7.7929377461265491430+02
9.739360766438018065D+06
1.819331998670494194D+00
2.169424162355176078D+01
1.885753194571055715D+02
4.474643508010343922D+02
5.430903102242293926D+03
2.059481936928183040D+04
Figure 46. Continued.
136
3.12977744121435648594E+02
I.653268819615301375D+02 
2.4499052653566023350+02 
1.1899567920503273480+02 
2.1433126121832414130+03 
2.9244527665203473770-03 
2.1152821683231283200+00 
4.3349340037697110530+00 
3.125395297778056118O+02
1.0061492780618058160+03
-4.0199544968637439270+02
8.8252889270252380530+02
2.4533013266945992650+04
1.2543061043450971-500+00
2.1342857257765893150+00
1.0973527351818170630+01
7.1769684520450630320+03
2.781792166908081522O-05 
1.4313195895848919370-03 
1.1747408602143863400-02 
9.4237822660506757880-01 
2.3716114463010808060+01 
-7.7057704294641909660+04 
2.018214735625751391O+08 
1.9855252595930897830-02 
2.4663386047684939250+00 
8.1803866318129736970+01 
3.7959200565884231790+02 
2.4917485706830576650+03 
5.7012277850562589260+04 
4.3476750254395662340+05 
-3GT001CGT391C
8.42999570573233378778E-04
1.0351262219100345740-03 
3.1509513577964530.900-03 
4.0997692978206763500-02 
8.8820792391493219590-01 
2.4010895233022150920+02 
4.4235407999739044500+04 
-2.0282700571782906350+08 
1.7573019966458918170+00 
5.4019603201826447150+00 
1.3980822164242446700+02 
3.969778083739700492O+02 
1.172996860918378343O+04 
1.3468147820709869850+05 
4.3519818406337345370+05 
I.
4.2960031754212272940+01 
1.3038754413689343790+00 
-5.4850742948332354840+02 
6.3709378844527256440+06 
1.7704737274829381870+00 
2.5620691470853891310+00 
1.1109072346148963770+01 
1.8799802743398523890+06
I.399449360722877848O-03 
1.6416004204058787330-03 
1.4514151520978417830-01 
5.317002636812340893O-01 
8.6183275291316975200+04 
3.6385429458747285390+05 
5.8805048774369715830+05 
2.3580802275281400050+00 
1.137916042757431345O+01 
2.4270250350855249980+02 
4.6469593931877139430+02 
5.6849265806544864970+04 
2.6572390157211301270+05 
6.2019627971760787480+05 
-2 6
2.7324304276277306513SE+02
1.5617483649513330230+02
4.9547372268967353650+03
2.8584466615883305000+02
7.7838725684164467470+01
2.858341495169046203O-03
1.7136224060601089050+00
1.0032645705042447130+01
3.1043758304139415840+01
8.0250734589119301750+03 -1.2188039451229226420+04
2.1124917674243302470+01
-2.9004944591815756020+01
8.4173181837846062990+04
1.5455735845919957860+00
2.6269694051641703240+00
1.0701826514185406980+01
5.4958611341160007210+04
8.41552877364927206030E-04
2.9324477898356918790-05 
1.5131597457534153790-03 
1.3015804244419530360-03 
1.9343137508618974460-02 
3.0058819605839396940+00 
I.3054020553714976240+03 
1.0236544555526942300+04 
1.5617147111654460430+08 
1.9854497691132311130-02 
2.5878178422434395810+00 
8.9420964772415199030+00 
2.6242465534884934190+02 
1.2343671067820312150+03 
1.4773044850261276680+04 
1.0748450137647499830+05 
6.7323000223326546260+05 
-4GT002AGT392A
1.0869183628724009680-03 
2.4976043517113766590-03 
1.8628011626329863990-03 
5.7463218795909463270-02 
3.0866951717226260850+00 
-8.6546522096864107710+02 
7.9361359940956410360+04 
-1.575334268721406125O+08 
1.8415674072963867660+00 
4.2853134203740785860+00 
1.2831313195218488010+01 
3.8946825959856095520+02 
1.3438406273771485640+03 
1.4813595284128386080+04' 
2.2788030577584638010+05 
6.7389690516088140430+05 
I.
2.20177781503156975162E+02
1.5334299491956741690+02 
-1.1337424806305393470+03 
6.1906238050969538730+02 
8.0631876443797688480+01 
1.2797038786930137800+04 
2.6891370423575039790-03 
2.0457005054009194110+00 
8.0920591315028709280+00 
2.0678317567295207980+01 
9.219693579084862222O+03
1.023260662629426164O+03
6.9306446303760046130+02
-1.4170532449340853280+02
7.0418537954657326240+01
1.3248999421426642940+00
2.0889905581289495150+00
8.9857604164882407540+00
3.2028890491373817720+01
1.9773129388126748300+02 
7.6187603663774323690+01 
6.0193381455366752340+06 
1.6475139350511790740+00 
5.2970315946850540230+00 
I.783766831197359082O+01 
3.9461761897403852780+06
1.1866534181968867580-03 
2.1293174030612647940-03 
1.2235700732917587690-02 
2.2002646726959117910-01 
1.7787465235403683570+00 
7.1917602007183453450+03 
8.3862253092658707460+05 
4.2609543113809363420+05 
2.032919689154692011O+00 
7.2687328445891279390+00 
1.7721091549876600620+02 
7.3682628430194111500+02 
1.5511310208981779510+03 
9.0754693824002686600+04 
4.5212223240590366910+05 
1.1423571857934275470+06 
-2 6
1.7015119021647301880+02 
1.8924158779865271280+01 
6.2416494870895014020+01 
6.8909409993101546020+01
1.9211515925114664160+00
2.8409808994294574740+00
1.2641734812466662640+01
5.6304544160761309970+01
Figure 46. Continued
137
8.39050201361497285067E-04
3.345672249978830877D-05 
5.446100361995422079D-03 
2.072081972291936122D-03 
4.013448422814709995D-01 
7.685394650151693652D+00 
1.397295661749917281D+04 
2.707494589161305339D+05
1 .824931134321452087D-03 
4.482048568296678791D-03 
4.54972394759873902ID-02 
8.285052595038885637D-01 
2.3583597050556461030+02 
3.8526706453043639160+04 
8.9343324304984636550+05
1.5030022089444381420+07 -9.5912362797264291440+06
1.9852426687611756870-02 
5.2632552625300362830+00 
1.6426216060709004200+01 
7.5925507295355278360+02 
3.764247589439823628O+03 
2.1486112541657514880+05 
2.2408377016633243180+06 
2.1412464965594399250+07 
-5GT002BGT3 92B
1.7550593259217452390+00 
8.5726983896387214390+00 
3.5889037319977207120+02 
8.0655140738780393410+02 
1.4263786260762994970+04 
5.9316720335386792430+05 
5.0584863114709854130+06 
2.6846268333910151850+07 
I.
2.33212776567026324415E+02
I.5299414460735035350+02 
3.2565735794866019860+02 
2.3314478611733502330+01 
6.8301716057848020470+01 
2.7433555016360898190-03 
1.8672691369632636040+00 
8.8955258211500176650+00 
3.0552473406546401420+01
1.0008691011699775800+04
3.4068929750995424750+01
6.3385397167198918920+01
7.3687392363851148500+02
1.5476493631996910140+00
2.8509241157061778750+00
1.2038205667102330750+01
5.2118828474229172800+02
8.39084788788901428922E-04
3.2182135086661849690-05 
4.7443247357915811590-03 
3.5780103626365829380-03 
2.0853144835420333590+00 
2.2845638170704132270+01 
1.8449294843958942690+04
1.0796297517001574690-03 
4.2372694392119293010-03 
6.7516866682606894650-02 
8.419199419823845953O-01 
3.9030420206726685710+02 
2.4225104132685916560+05
4.6448935960374594430+06 -1.8612512617159653750+06
2
2
2
I
1 
6 
6
2
1 
4
2 
4 
I 
6
-2  I
-9
4
8
I
I
7
I
9
I
3. 
3
4.
1. 
4.
2.
-2.1856190587187005620+11 
1.9853065556187481090-02 
3.6931507364784086360+00 
2.2443585483480681830+01 
8.1519410838224145270+02 
8.8604164512381826170+03 
4.3144191367892330160+05 
5.4531873485321948540+06 
1.8119433569013459140+07 
-6GT002CGT392C
1.8846893122797775650+11 -I 
1.6681373509157113030+00 I 
6.6226951551049033640+00 
4.1951915861599127310+02 
1.2881278701744030000+03 
1.9213994600743156300+04 
1.2029740745803923930+06 
7.6616340558706353190+06 
1.8705641094308915080+07 
I.
9, 
6. 
4.
1.
2. 
I. 
I.
-2 (
2 .38428075051592937683E+02
1.5629549262023844850+02
1.5104145384349836600+04
-1.3842893753893300610+02
7.2180769215806682480+01
2.7492069186787277320-03
1.6984339661205038460+00
8.4209633936186567560+00
3.1179644538158888700+01
1.2422949426016137070+04 -2.
3.3815330156323950560+01 
6.668169235828113273O+01 
8.8754864041939477200+02 
1.5787265010572736620+00 
2.8551354925706502460+00 
1.2086977363596657310+01 
6.1432240119109611950+02
8.39124355762650754880E-04
3.2230021740363911860-05 
1.997204523683885897O-03 
2.0090315367938778920-03 
1.5390875918277028560+00 
6.1303398675024845940+02 
3.5276714219630160190+04 
1.5676523304940540770+05 
I
—8
1.9853041548219403990-02
2.8553315742970628580+00
1.1594477828320990610+01
1.0851096527589688210+03
2.7183571461350078610+04
5.6697081501736126670+05
1.6443348372643347830+06
1.0392434193667609010+07
1.426841444457575733O-03 I. 
3.1906270422291114540-03 6. 
5.7156718095517957710-02 I. 
9.2947626068541759900+00 7. 
4.5508001459554443500+03 I. 
1.0924827969933577330+05 3. 
5.0542519895966749170+06 -3.
9.2577629858389049050+10 -1.2845728432862173800+10 
8107753474688501360+09
1.9698216002974028860+00 2. 
4.5652700371405047220+00 9. 
3.2805190632063979450+02 5. 
1.6617727316040491130+03 6. 
9.8843200201908923190+04 2. 
8.6779588560998594040+05 I. 
3.6574127665491899820+06 4. 
1.0402728958853091110+07 I.
I.1286508246563949620+07
.7142359535255377800-03
.9686852855093579500-03
.5705589480464727250-01
.9176079540959140250+01
.5319320701324842280+03
.7985802235523558920+04
.4960036719306551090+05
.6223078662985611630+00
.1571217034621940510+01
.9829140091489668410+02
.3144501041523791400+03
.7229740461283957300+04
.1042315381209675400+06
.5550506091531781710+06
.5927538032942463810+03
.1905937481175509160+02
.0344690052234092550+01
.3905095131100364600+04
.5992873419360305040+00
.3066042312692631720+00
.9716647350363850410+01
.8414997961683293400+03
.2234909193828333300-03 
0443849709261753100-03 
.8394953692096887160-01 
5192560796724592360+01 
1835328188287183590+04 
0273261504565890210+05 
0664791156043748470+11 
7655839628889772420+11 
9079578050730287010+00 
5486957820101814320+00 
0172627172262187400+02 
3902548087164594790+03 
4359153185792697700+05 
3082633365193682840+06 
8101502186332554090+07 
8724170938147344160+07
6763768257296094820+04
9405735716945244460+02
7144830731491104810+01
8312560433208102040+04
6620983625157836620+00
6080824387320934440+00
9098149205273395430+01
2646103243562018860+04
3738398884081777430-03
6906013836810679420-03
7873862645276226630-01
2359943515528614810+01
4323714718939104390+04
2038497334277586920+05
425473663962425082O+06
0766038974207980630+09
0157299471767684000+00
5670145094880236770+00
0722910514118201770+02
4654271397523710900+03
8328138813770514390+05
4159440923027529790+06
2287626627391459190+06
1275338874313460430+07
Figure 46. Continued
138
c Q MATRIX 
0.50087070 
0.00000000 
0.34049665 
0.00000000 
0.35677436 
0.00000000 
0.50087517 
0.00000000 
0.34049490 
0.00000000 
0.35677883 
0.00000000
BY ROWS (IMAGINARY PART 
-0.46167319 -0.49313081
0.00000000
-0.02109095
0.00000000
0.53295208
0.00000000
-0.46168710
0.00000000
-0.02108988
O.OO0OOOOO
0.53297823
0.00000000
0.00000000
-0.44756805
0.00000000
-0.18697741
0.00000000
0.49313768
0.00000000
0.44755453
0.00000000
0.18698441
0.00000000
= 0)
0.24515321
0.00000000
-0.44697417
0.00000000
0.48533423
0.00000000
-0.24513653
0.00000000
0.44693392
0.00000000
-0.48531000
0.00000000
0.47310880
0.00000000
-0.25127029
0.00000000
-0.44765855
0.00000000
-0.47333592
0.00000000
0.25174401
0.00000000
0.44741519
0.00000000
-0.30648209
0.00000000
0.58002629
0.00000000
-0.26281078
0.00000000
-0.30604273
0.00000000
0.57993411
0.00000000
-0.26322140
0.00000000
C ========================EQUIVALENTS FOR EXTERNAL SYSTEM===================
C 1 2 3 4 5 6 7 8  
C 345678901234567890123456789012345678901234567890123456789012345678901234567890
C
-1CS230ABV230A
-2CS230BBV230B
-3CS230CBV230C
C
-1BV230AGA230A
-2BV230BGA230B
-3BV230CGA230C
C
-1HS230AGA230A 
-2HS230BGA230B 
-3HS230CGA230C 
C 51CS230A 
52CS230B 
53CS230C
51BV230A 
52BV230B 
53BV230C 
51HS230A 
52HS230B 
53HS230C 
C
51GA230A
52GA230B
53GA230C
51TF500AHS500A
52TF500BHS500B
53TF500CHS500C
51TF5 0 0ADW5 00A
52TF500BDW500B
53TF500CDW500C
51TF5 0 0ABE5 0 OA
52TF500BBE500B
53TF500CBE500C
51BE500A
52BE500B
53BE500C
51DW500A
52DW500B
53DW500C
C ===========
368102.80883.3763131.44
08191.753035.6271131.44
! ! ! !
184051.40446.752683.5
04095.3765211.25483.5
! ! ! !
184051.40446.7526135.0
04095.3765211.254135.0
.00529 5.0202100 
.04232 21.511785
! ! ! !
2.6609 32.083850 
2.4202 45.665925 '
26.857
24.366
246.13841
381.87981
! ! I
4.3748 41.833320 
3.7533 50.699360
17.134
1.6922
116.2794
32.5802
28.557 193.7991 
2.8204 54.3003
32.91 218.60
3.25 61.25
176.25
55.75
44.38
34.25
=END OF EXTERNAL EQUIVALENTS
Figure 46 Continued
O
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d 
O
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139
C ==================BEGIN THEVENIN IMPEDANCES ================
C 345678-1-2345678-2-2345678-3-2345678-4-2345678-5-2345678-6-2345678-7-2345678-8 
BTHEVABE5 OOA 55.75
BTHEVBBE5 0 0BBTHEVABE5 0 OA 
BTHEVCBE50 0CBTHEVABE5 OOA 
DTHEVADW5 OOA 34.25
DTHEVBDW5 0 0BDTHEVADW5 OOA 
DTHEVCDW5 0 0CDTHEVADW5 0 OA 
HTHEVAHS230A .3703 32.639
HTHEVBHS23 0BHTHEVAHS23 OA 
HTHEVCHS230CHTHEVAHS230A 
GTHEVAGA23OA 3.703 44.066
GTHEVBGA23 0BGTHEVAGA23 OA 
GTHEVCGA23 0CGTHEVAGA2 3 OA 
BVTHVABV23OA 2.486 43.219
BVTHVBBV2 3 OBBVTHVAB V2 3 OA 
BVTHVCBV23 0CBVTHVABV23 OA
C = ============== =====:LINE CHARGING DATA FOR TF-HS, DW, BE=
TF500A 974.86
TF500B 974.86
TF500C 974.86
HS500A 189.28
HS500B 189.28
HS500C 189.28
DW500A 315.46
DW500B 315.46
DW500C 315.46
BE500A 470.12
BE500B 470.12
BE500C 470.12
=END OF BRANCH DATA=
LANK CARD TO END BRANCH DATA
========================BEGIN SWITCH DATA=============;
=============CIRCUIT BREAKERS========================;
COLSTRIP -
Bus— >Bus— x  Tclose<----Topen<
CBOO1ACS5 0 OA -.020 12.0
CB001BCS500B -.020 12.0
CB001CCS500C -.020 12.0
CB002ACS500A -.020 12.0
CB002BCS500B -.020 12.0
CB002CCS500C -.020 12.0
BV5OOABCSWlA -.020 12.0
BV5OOBBCSWlB -.020 12.0
BV5 0 0CBCSW1C -.020 12.0
BV5 0 0ABCSW2 A -.020 12.0
BV500BBCSW2B -.020 12.0
BV500CBCSW2C -.020 12.0
BROADVIEW:
Ie
C ============================BROADVIEW - GARRISON========================
BG001ABV500A -.020 0.298 I
BG001BBV500B -.020 0.298 0
BG001CBV500C -.020 0.298 0
C BG001ABV500A -.020 12.00 I
C BG001BBV500B -.020 12.00 0
C BG001CBV500C -.020 12.00 0
BV500ABG0OlA 1.112 12.0 0
BV500BBG0OlB 1.112 12.0 0
BV5OOCBGOOlC 1.112 12.0 0
BG002ABV500A -.020 12.0 0
BG002BBV500B -.020 12.0 0
BG002CBV500C -.020 12.0 0
GA500AGBSW1A -.020 0.298 0
GA500BGBSW1B -.020 0.298 0
GA5OOCGBSWlC -.020 0.298 0
C GA500AGBSW1A -.020 12.00 0
C GA500BGBSW1B -.020 12.00 0
C GA5OOCGBSWlC -.020 12.00 0
GBSW1AGA500A 1.112 12.0 0
GBSW1BGA5 00B 1.112 12.0 0
GBSW1CGA5 OOC 1.112 12.0 0
GA500AGBSW2A -.020 12.0 0
GA500BGBSW2B -.020 12.0 0
GA500CGBSW2C -.020 12.0 0
Figure 46. Continued.
O
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O
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C140
- g a r r i s o n  - t a f t ===
GT391ATF500A - .020 12.0
GT3 91BTF5 OOB - .020 12.0
GT3 91CTF5 00C - .020 12.0
GT392ATF500A - .020 12.0
GT392BTF500B - .020 12.0
GT392CTF500C -..020 12.0
GA500AGTSW1A -..020 12.0
GA500BGTSW1B -..020 12.0
GA5OOCGTSWlC -..020 12.0
GA5 0 0AGTSW2A -..020 12.0
GA5 0 0BGTSW2 B -..020 12.0
GA500CGTSW2C -..020 12.0
C ======================== FAULT SWITCHES ==================================
BGFFlA .220 1.16
BGFFlB .220 1.16
BGFFlC .220 1.16
C
C ======================= TACS CONTROLLED SWITCHES FOR BRAKE ===============
C Type 11 Switches are TACS Controlled Switches. These are SCR'S.
C 345678-1-2345678-2-2345678-3-2345678-4-2345678-5-2345678-6-2345678-7-2
IlTMP AlBRAKEA BRKOPN
11TMP_A2_NGH_A BRKOPN
IlTMP BlBRAKEB BRKOPN
11TMP_B2_NGH_B BRKOPN
11TMP_C1BRAKEC BRKOPN
11TMP_C2_NGH C BRKOPN
C ========================END SWITCH DATA===================================
BLANK CARD TO END SWITCH DATA
I
C ============
C I
C 345678901234. thevenins
BEGIN SOURCE DATA================================
3 4 5 6 7 8
. . .45678901234567890123456789012345678901234567890
14DTHEVA 455000. 60. -20.00 -I.
I4DTHEVB 455000. 60. -140.0 -I.
14 DTHEVC 455000. 60. 100.0 -I.
I4BTHEVA 455000. 60. -32.53 -I.
14BTHEVB 455000. 60. -152.53 -I.
I4BTHEVC 455000. 60. 87.47 -I.
14HS230A 165348. 60. -10.0 -I.
14HS230B 165348. 60. -130.0 -I.
14HS230C 165348. 60. 110.00 -I.
I4BVTHVA 177372. 60. 16.00 -I.
I4BVTHVB 177372. 60. -104.00 -I.
14BVTHVC 177372. 60. 136.00 -I.
C CS230A 197184. 60. 26.17 -I.
C CS230B 197184. 60. -93.83 -I.
C CS230C 197184. 60. 146.17 -I.
14GTHEVA 180175. 60. 7.0 -I.
14GTHEVB 180175. 60. -113.0 -I.
14GTHEVC 180175. 60. 127.00 -I.
C ....... .......Voltage Contr. Source data for brake ...
60VSRCEA I 
60VSRCEB I 
6OVSRCEC I
-I.
-I.
-I.
9999.
9999.
9999.
G Dynamic synchronous machines.
C Bus— > <----- Volt<----- Freq<---- Angle
59CS221A 17941.37 60.0 -1.90
CS221B 
CS221C
TOLERANCES I .OE-12
PARAMETER FITTING I .0
3 3 0 2 377.0 22.0 -971.3 1150.0 1788.4
-971.3 1150.0 1788.4
0.0052 0.175 1.72 1.71 0.277 0.373 0.253 0.253
5.50 1.80 0.03 0.07 0.110 2187.0 271.0
01 0.550 0.040101 0.000 0.000 43.05367
02 0.450 0.191576 0.000 0.000 51.00225
03 0.000 0.116460 0.000 0.000 00.00000
BLANK card terminates mass cards 
BLANK card terminates output requests 
FINISH
Figure 46. Continued.
H
O
O
O
O
O
O
O
O
O
O
O
141
C Dynamic synchronous machine.
C Bus--> <----- Volt<----- Freq<---- Angle
59CS222A 17937.78 60.0 -1.90
CS222B 
CS222C
TOLERANCES 1.0E-12
PARAMETER FITTING I .0
3 3 0 2 377.0 22.0 -971.3 1150.0 1788.4
-971.3 1150.0 1788.4
0.0052 0.175 1.72 1.71 0.277 0.373 0.253 0.253
5.50 1.80 0.03 0.07 0.110 2187.0 271.0
01 0.550 0.040101 0.000 0.000 43.05367
02 0.450 0.191576 0.000 0.000 51.00225
03 0.000 0.116460 0.000 0.000 00.00000
C I 2 3 4 5 6 7
C 345678901234567890123'456789012345678901234567890123456789012345678901234567890 
BLANK card terminates mass cards 
31 
41
BLANK card terminates output requests 
FINISH
C SCOURGE DATA FOR COLSTRIP UNIT THREE AND FOUR (PARALLEL) 
C Dynamic synchronous machine.
C Bus— > <----- Volt<----- Freq<----Angle
59CS261A 21727.79 60.0 -1.90
CS261B 
CS261C
PARAMETER FITTING I .0
6 5 6 2 1638.0 26.0 -5800. 6200.0 11200.0
-5800. 6200.0 11200.0
0.00132 0.1284 1.2356 1.2221 0.2197 0.3482 0.1776 0.1742
4.777 0.531 0.041 0.067 0.0861 2329.0 640.0
01 0.272 0.063598 941.4 201.50 53.50 0.000
02 0.275 0.119634 1171. 809.6 73.34 0.000
03 0.2265 0.598438 963.2 809.6 73.34 0.000
04 0.2265 0.605408 963.2 724.2 110.5 0.000
05 0.000 0.504274 0.000 144.4 58.34 0.000
06 0.000 0.024816 1056.4 0.000 0.000 0.000
C .1 2 3 4 5 6 7
C 345678901234567890123456789012345678901234567890123456789012345678901234567890 
BLANK card terminates mass cards 
31 
41
BLANK card terminates output request.
C TAGS input cards.
C ------------------------------------------
C The following card enters the gen velocity into array ETACS to be picked up 
C by the control subroutine.
C ------------------------------------------
C BUS--X---><KI
7 4GENVEL 11
C col: (1-2) KK - either 71, 72 , 73 or 74 
FINISH
BLANK CARD TO END MACHINE DATA 
CS500A
BLANK CARD TO END NODE-VOLTAGE OUTPUT SPECIFICATION 
BLANK CARD TO END BATCH MODE CONTROL CARDS 
BEGIN NEW DATA CASE 
BLANK
Figure 46. Continued
142
APPENDIX C
EMTP INTERFACE AND CONTROLLER CODE
143
This appendix contains the information necessary to 
interface the dynamic brake to the EMTP digital simulation 
package. Figure 47 shows the FORTRAN source code used to 
obtain the control values at each simulation time step. The 
subroutine INJECT, shown in Figure 47, is called from the 
modified EMTP subroutine CSUP. CSUP is is shown in Figure 
48. The procedure to implement the brake in the EMTP would 
be to first modify CSUP by adding the subroutine call statement. 
Then CSUP and INJECT would be compiled and linked with the 
original EMTP library. The data files in Appendices A and 
B could then be simulated following the procedures spelled 
out in an EMTP User's Manual supplied with the EMTP code.
Q * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *
C *** 
C Controller subroutine to be called from CSUPB in EMTP. ***  
C The called variables are: *** 
C *** 
C !STEP - the step number in the time domain simulation *** 
C DELTAT - the step size
C V a -  the "A" phase terminal voltage for the gen being controlled 
C V b -  the "B" phase terminal voltage for the gen being controlled 
C V c -  the "C" phase terminal voltage for the gen being controlled
C
C The returned variables are: ***
C ***
C VBa - the set point "A" phase voltage for the Type 60 source ***
C VBa - the set point “A" phase voltage for the Type 60 source ***
C VBa - the set point "A" phase voltage for the Type 60 source ***
C brake - the thyristor gate signal for the TACS switches ***
C ***
Q **********************************************************************
Q **********************************************************************
SUBROUTINE INJECT (ISTEP, DELTAT, V a , V b , V c , VBa, VBb, VBc, brake)
*CALL IMPLIDP GPC B
IMPLICIT DOUBLE PRECISION (A-H1 O-Z) IMPLIDP
*END IMPLIDP .......
DIMENSION Yold(-2:0),Ynew(-2:0)
*CALL SYNCOMB
C THIS DECK CONTAINS S.M. STORAGE USED BY TAGS MODULES. SYNCOMB
COMMON /SMTACS/ ETAC(20) ■ SYNCOMB
COMMON /SMTACS/ ISMTAC(20), NTOTAC, LBSTAC SYNCOMB
*END SYNCOMB .......
DATA samplefreq /200.D0/
C Decide whether this is a sample point or not. This is useful 
C for digital controller implementation. If controller is continuous 
C then comment out the IF statement.
TIMEX=ISTEP*DELTAT*10000 .DO 
samplerate=!.D0/samplefreq*10000-DO 
Itime=IDNINT(TIMEX)
Irate=IDNINT(samplerate)
Itimemodulo=MOD(!time,irate) 
c IF (itimemodulo .NE. 00 GO TO 777
Figure 47. Controller subroutine INJECT.
144
C Update the filter coefficients.
C DELTAVGEN is the current generator delta omega.
Yold(-2)=Yoldf-I)
Yoldf-D=Yold(O)
deltavgen=etac(I )-376.991118431D0 
YoldfO)=deltavgen 
Ynewf-2)=Ynewf-D 
Ynew(-I)=Ynew(O )
C Implement the filter. This is a high-pass digital filter 
C with sample rate = 10,000 and cutoff = 2 hertz.
YnewfO)=1.D0*Yold(0)-I.99999967967526D0*Yold(-I)
&+0.99999967967526D0*Yold(-2)+1.99835970522999D0*Ynew(-I)
&-0.9983603457043ODO *Ynewf-2)
C The control action U is the gain times the filtered signal.
U= 9.d0*Ynew(O)
C The following lines implement a "dead-zone"scheme. If the filtered 
C signal stays below .2 for three time steps in a row the control 
C action is always zero.
C Ysav=O.dO
C DO I=-2,1,0
C IF(YnewfI) .GT. 0.2D0) Ysav=I.DO
C END DO
C if(Ysav.EQ.0 .DO) U=O.dO
C Uncomment this to run the uncontrolled case, 
c U= 0 .DO
C The brake is on when "brake“=l. The IF statements are for implemeting 
C bang-bang control. These can be commented out for classical control.
brake=0.DO
IF (U .LT- 0.1D0) U=O-DO .
C IF (U .GT. I.DO) U = D D O
IF (U .GT, 0.D0) brake=I.DO
C These set the voltage to the Type 60 sources.
VBa=Va*(1.D0-DSQRT(U))
VBb=Vb*(I.DO-DSQRT(U))
VBc=Vc*(I.DO-DSQRT(U))
c writef*,*) ' hit point',istep,deltavgen,u
777 RETURN 
END
Figure 47. Continued.
Figure 48 shows some of the EMTP subroutine CSUP. Much 
of the subroutine has been deleted for brevity. The CALL 
statement for the controller subroutine shown in Figure 47 
can be found at the end of CSUP -- just before the RETURN 
statement.
SUBROUTINE CSUPf L ) CSUPB I
*CALL IMPLIDP CSUPB 2
IMPLICIT DOUBLE PRECISION (A-H, O-Z) IMPLIDP I
*END IMPLIDP ........  0
GPC B 
GPC B
Figure 48. Modifications to EMTP Subroutine CSUP
145
Common block declarations of Subroutine CSUP deleted
*CALL TACSARB CSUPB
C 000000 DEFINITION OF TABLE NAMES 000000TACSARB
DIMENSION ISBLK (I), INSUP (I), JOUT (I), ICOLCS(I) , ILNTAB(I) TACSARB
DIMENSION KSUS (I), IUTY (I), IVARB (I)
DIMENSION RSBLK (I), UDl (I), XTCS (I), ATCS (I), XAR (I)
DIMENSION PARSUP(I) , AWKCS (I)
C EQUIVALENCING OF SCALARS WHICH ARE TO BE CARRIED BETWEEN MODULES.
EQUIVALENCE (KONSCE, SPTACS( I)), 
EQUIVALENCE (KONTOT, SPTACS( 3)), 
EQUIVALENCE (KCOLCS, SPTACS( 5)), 
EQUIVALENCE (KATCS , SPTACS( 7) ) , 
EQUIVALENCE (KPRSUP, SPTACS( 9)) , 
EQUIVALENCE (KALIU , SPTACS( 11)), 
EQUIVALENCE (KIUTY , SPTACS( 13)), 
EQUIVALENCE (KAWKCS, SPTACS( 15)), 
EQUIVALENCE (KXTCS , SPTACS( 17)), 
EQUIVALENCE (KISBLK, SPTACSf 19)), 
EQUIVALENCE (KKSUS , SPTACS( 21)), 
EQUIVALENCE (KINSUP, SPTACS( 23))
(KONCUR, SPTACS( 
(KOFSCE, SPTACS( 
(KSPVAR, SPTACS(
(KONSUP, 
(KIVARB, 
(KJOUT , 
(KUDl , 
(KXAR
SPTACS( 
SPTACS(
2) ) 
4) ) 
6) ) 
8) ) 
10) )
SPTACS( 12))
SPTACS( 
SPTACS(
(KLNTAB, SPTACS( 
(KRSBLK, SPTACS( 
(KALKSU, SPTACS(
14) ) 
16) ) 
18) ) 
20) ) 
22) )
(
*END
EQUIVALENCE 
EQUIVALENCE 
EQUIVALENCE 
EQUIVALENCE 
EQUIVALENCE 
EQUIVALENCE 
EQUIVALENCE 
EQUIVALENCE 
EQUIVALENCE 
EQUIVALENCE 
TACSARB
DIMENSION ARG( 50), ACC( 20), AMX( 20) 
DIMENSION IOP( 50), IFL( 20), IDN( 20)
SPTACS(I ) 
SPTACS(I) 
SPTACS(I) 
SPTACS(I), 
SPTACS(I), 
( NUK 
( NSU 
( NSUP ,
( KPAR ,
( IOUTCS,
ISBLK (I) 
ILNTAB(I ) 
INSUP (I) 
RSBLK (I) 
ATCS (I ) 
LSTAT(Sl) 
LSTAT(53) 
LSTAT(55) 
LSTAT(57) 
LSTAT(59)
KSUS (I), 
ICOLCS(I), 
IVARB (I),
IUTY
JOUT
)
UDl
XAR
(I)
CD-,
( IA 
( NIU 
( KARG 
( KXIC 
( NSUDV
(I )
(I)
PARSUP(I) ) 
XTCS (I) ) 
AWKCS (I) ) 
LSTAT(52) 
LSTAT(54) 
LSTAT(56) 
LSTAT(58) 
LSTAT(60)
TACSARB
TACSARB
TACSARB
TACSARB
TACSARB
TACSARB
TACSARB
TACSARB
TACSARB
TACSARB
TACSARB
TACSARB
TACSARB
TACSARB
TACSARB
TACSARB
TACSARB
TACSARB
TACSARB
TACSARB
TACSARB
TACSARB
TACSARB
TACSARB
TACSARB
TACSARB
CSUPB
CSUPB
5
1
2
3
4
5
6
7
8 
9
10
11
12
13
14
15
16
17
18
19
20 
21 
22
23
24
25
26
27
28 
0 
6 
7
C 000 B = THE ARGUMENT AND LATER THE VALUE OF THE FUNCTION, CSUPB 8
C 000 BEFORE IT IS AFFECTED BY THE ALGEBRAIC OPERATION CSUPB 9
C 000 WHICH WILL UPDATE 'A' . CSUPB 10
C 000 A = INTERMEDIATE VALUES OF THE SUPPLEMENTAL CSUPB 11
C 000 VARIABLE OR DEVICE . CSUPB 12
Much of Subroutine CSUP deleted
C ---M IN/MAX TRACKING, CONTROLLED ACCUMULATOR OR COUHTER --- CSUPB 654
664 NDX4 = KXTCS + IVARB( NI + 3 ) CSUPB 655
IF ( NDX4 .EQ. KXTCS ) GO TO 6641 CSUPB 656
IF ( XTCS( NDX4) .LE. FLZERO ) GO TO 6641 CSUPB 657
A = PARSUP( NN + 2 ) CSUPB 658
GO TO 6643 CSUPB 659
6641 NDX5 = KXTCS + IVARB( NI + 4 ) CSUPB 660
Figure 48. Continued.
146
IF ( NDX5 .EQ. KXTCS ) GO TO 6642 CSUPB 661
IF ( XTCS( NDX5) .LE. FLZERO ) GO TO 6642 CSUPB 662
A = PARSUP(NN) CSUPB 663
GO TO 11 CSUPB 664
6642 IF ( N2 .EQ. 16 ) GO TO 665 CSUPB 665
A = PARSUP(NN) CSUPB 666
RDEVl = PARSUP( NN + I ) CSUPB 667
IF ( RDEVl .EQ. -1.0 .AND. B .LT. A ) A = B CSUPB 668
IF ( RDEVl .EQ. +1.0 .AND. B .GT. A ) A = B CSUPB 669
GO TO 6643 CSUPB 670
665 A = PARSUP(NN) + B CSUPB 671
6643 PARSUP(NN) = A CSUPB 672
GO TO 11 CSUPB 673
■666 IVARB(Nl+4) = IVARB(Nl+4) + I CSUPB 674
K = IVARB( NI + 3 ) CSUPB 675
IF ( IVARB(Nl+4) .GT. K ) IVARB(Nl+4) = I CSUPB 676
NDXl = NN + IVARB( NI + 4 ) CSUPB 677
PARSUP(NDXl) = B * B CSUPB . 678
DO 6666 JI = I, K CSUPB 679
6666 A = A +  PARSUP( NN + JI ) CSUPB 680
A = SQRTZ ( A * PARSUP (NN.) ) CSUPB 681
GO TO 11 CSUPB 682
667 J = IVARBt NI + I ) CSUPB 683
K = IVARB( NI + 2 ) CSUPB 684
B = 0.0 CSUPB 685
DO 1166 MJ = J, K CSUPB 686
N = KKSUS + MJ CSUPB 687
IF ( KSUS(N) .EQ. 9 ) GO TO 1166 CSUPB 688
M = KALKSU + MJ CSUPB 689
BB = XTCS( KXTCS + KSUS(M) ) * KSUS(N) CSUPB 690
NJ = MJ CSUPB 691
1144 NJ = NJ - I CSUPB 692
N = N - I CSUPB 693
M = M - I CSUPB 694
IF ( NJ .LT. J ) GO TO 1155 CSUPB 695
IF ( KSUS(N) .NE. 9 ) GO TO 1155 CSUPB 696
BB = BB * XTCS( KXTCS + KSUS(M) ) CSUPB 697
GO TO 1144 CSUPB 698
1155 B = B + BB CSUPB 699
1166 CONTINUE CSUPB 700
A = B *  PARSUP( NN + 3 ) CSUPB 701
AA = A * PARSUP( NN ) CSUPB 702
IF ( AA .GE. PARSUP(NN+I) ) GO TO 6677 CSUPB 703
A = PARSUP(NN+I) / PARSUP(NN) CSUPB 704
GO TO 11 CSUPB 705
6677 IF ( AA .LE. PARSUP(NN+2) ) GO TO 11 CSUPB 706
A = PARSUP(NN+2) / PARSUP(NN) CSUPB 707
11 NDXl = NNN + I CSUPB 708
XTCS(NDXl) = A CSUPB 709
10 I = INSUP(KINSUP+I) CSUPB 710
IF ( I .GT. 0 ) GO TO 1234 CSUPB 711
Q*************************************************************************
C Call the controller subroutine to compute the control action.
C XTCS holds the voltages specified by the TACS Supplemental Variables. 
C Generator Delta Omega is brought in through the common block ETACS.
Q*************************************************************************
call inject(!STEP,DELTAT,xtcs(nnn+1),xtcs(nnn+2),xtcs(nnn+3) 
Sxtcs(nnn+4),xtcs(nnn+5),xtcs(nnn+6),xtcs(nnn+7))
9999 RETURN 
END
MKD21491 I 
MKD21491 2 
CSUPB 712 
CSUPB 713
Figure 48. ■ Continued
147
APPENDIX D
ADAMS PREDICTOR-CORRECTOR TO IMPLEMENT A PASC SYSTEM
148
This appendix contains the source code to implement a 
fourth-order Adams Predictor-Corrector on an experimental 
PASC system. Much of the program is specific to the PASC 
system. This is particularly true with the INPUT subroutine 
which initializes the "A" and "B" matrices for the state-space 
formulation x=Ax+Bu. The generic implementation of the 
integration algorithm can be easily gleaned from the 
information contained herein. Figure 49 shows the entire 
program while Figure 50 shows an example of a file that can 
be used to execute the program for typical PASC data.
Q * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *
c* * ***
Program A _ M O U L T O N  is a 4-step, pre dictor -corre ctor integration * 
routine u si ng the fourth order A D A M S -BASHFORTH p r e d i c t o r  *** 
and the fourth ord er A D A MS-MOU LTON corrector. Fun ction *** 
calls are mad e to A PR ED and A C O R R  to advance to the next *** 
integration step. Since the global truncation err or ***
inherent in this scheme is on the order of h A4 , a step *** 
size of smaller than 10e-02 should not be taken w i t h  ***
single p re cision math. ***
C** 
C** 
C* *
C* * 
C**
C** 
C* * 
C* *
C* *
C* *
C**
c* *
c** 
c* *
c* *
M u l ti-ste p methods require mor e than one starting v al ue to * 
proceed. Usually, a Runge-Kutta m e t h o d  w ou ld be u s e d  to * 
get the first three steps. In some cases, however, it can 1 
be ass umed that the system has b e e n  at rest for some time * 
and the conditions at the first four steps are known (first 
three steps and the initial c o n d i t i o n ) . *Q** ***
Q * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *
PROGRAM A_M O U L T O N
C
C -
C
C
C
C
C
C
C
C
C
C - 
C 
C 
C
Subroutines called,
» I N P U T  - allows inputs to the r o u t i n e . (step size, etc.)
▻▻INITIAL - dev elops the four initial states required for the 
algorithm. One state is initial condition, and 
the rest are usually found w i t h  three steps of 
R u n g e - K u t t a .
▻▻S TATE - evaluates f = y ' = di/dt for use in the function 
calls. n n
▻ ▻ O UTPUT - sets up and records the output data.
Functions called,
▻▻APRED
▻▻ACORR
C - Declare variables
IMPLICIT R E A L *8 (A-H), (O-Z)
R E A L *8 T N O W ,T FI N,h,h O V E R 2 4 ,PERIOD,D C V OLTS,Linv,Linv R,Cur 
INTEGER N_p rnt_va r 
C H A R A C T E R * 20 FILENAME
Figure 49. Adams Predictor-Corrector for Experimental PASC
System.
149
c
C - D i m ension arrays 
C
D I M E N S I O N  Y n p l u s l (100),Yn (IOO),F n p l u s l (100), F n (IOO),F n _ l (100),
& F n _ 2 (100), F n _ 3 (100)
DIM E N S I O N  Cur(IOO) ,
C
C - Com mon buf fer caries some integration variables that sho uld be 
C de f i n e d  in input 
C
C O M M O N / IN T V A R S / T N O W ,T F I N ,T N R M L IZ D ,h ,hOVER2 4 
C O M M O N /INTVARS 2/ N_s t _ e q s ,N_btw_ou t ,N_j?rnt_var(50)
COM MON/STVARS/ PERIOD,V s ,L i n v (50,50), L i n v R (50,50)
C O M M O N / P A R A M S / a,R S l (50),R S 2 p (50),R S 2 s (50),Rl,S U P P L Y (50)
C O M M O N /P A R AMSI / L m , Ls
COM MON/PA RAMS2/ RSl closed ,RS2popen
C
C - Subroutine INPUT establishes integration parameters such as step 
C size and any input variables used.
C
h = 0 .DO 
CALL INPUT
C
C - Sub routin e INITIAL steps the integration through its first three 
C steps u si ng a single step m e t h o d  such as R u n g e - K u t t a . It may 
C also be p o s sible to assume that the system has been at rest
C for some time and use an assumed v al ue for the first three
C steps.
C
CALL I N I T I A L (Yn,F n _ l ,F n _ 2 ,F n _ 3 )
C
C - Initialize TNO W 
C
T N O W  = 0 .DO 
C
C - Call OUT P U T  to initialize the output file 
C
W R I T E (*,*) ' Input the name of the O U T PUT file. <fid.ext>'
R E A D (*,101) FILENAME
O P E N (UNIT=I,F I L E=FILE NAME,S T A T U S = 'N E W ' )
101 FORMAT(A20)
C
C - Sub routin e STATE evaluates the function at the current step. It 
C returns a val ue for Fn(I) from the values of Y n (I thru N_st_eqs)
C This par ticula r call evaluates Fn u si ng the c o r rected Yn for use
C in p re dictin g the value of Y n p l u s l .
C
C - B eg inning of integration loop.
C
111 CALL S T A T E ( F n zYn)
Figure 49. Continued.
n
o
n
 
o
o
o
o
 
o
o
o
o
o
o
 
o
o
o
o
 
o
o
o
o
150
- Sub routin e OUT P U T  provides a format for storage of the desired 
output at integration step n.
Tprnt = T N O W / (N_btw_out*h) - N I N T ( T N O W / ( N _ b t w _ o u t * h ) ) 
I F ( A B S (Tprnt) .LE. 1.0D-08) CALL O U T P U T ( Y n zFn)
- Fun ction A P R E D  pre dicts the value of Ynplusl u si ng an explicit 
algorithm.
DO 1=1,N_s t_eqs
Ynplusl(I) = A P R E D ( Y n d )  ,Fn(I) ,Fn_l(I) , Fn_2 (I) ,Fn_3 (I) ) 
END DO
- Sub routin e STATE evaluates the function at the current step. It
returns a v al ue for Fn(I) from the values of Y n (I thru N _ s t ^ e q s ) . 
This p ar ticula r call evaluates Fnplusl using the p r e d i c t e d  
Ynplusl for use in correcting the val ue of Y n p l u s l .■
CALL S T A T E (Fnplusl,Y n p l u s l )
- Fun ction A C O R R  cor rects- the value of Ynplusl using an implicit 
algorithm.
DO I=l.,N_st_eqs
Ynplusl(I) = A C O R R ( Y n d )  ,Fnplusl (I) ,Fn(I) ,Fn_l (I) ,Fn_2 (I) ) 
END DO
- M o v e  to the next integration s t e p .
DO I=l,N_st_eqs 
F n _ 3 (I) = F n _ 2 (I ) 
Fn_2 (I) = Fn_l.(I) 
FnJL(I) = Fn(I)
Yn(I) = Y n p l u s l (I) 
END DO
TN O W  = TNO W + h
- End of - integration loop
I F (TNOW .LE. TFIN) GO TO 777
STOP 'INTEGRATION COMPLETED' 
END
Figure 49. Continued.
151
c***********************************************************************
c** ***
C** A P R E D  is a fourth order A D A MS-BAS HFORTH p re dictor to be us e d  *** 
C** wi t h  a fourth order ADA M S - M O U L T O N  corrector A M O U L T . ***
C** The p r e d i c t o r  is of the form ***
C * * * * *
C** Y  =Y + h / 2 4 *(55f -59f +37f -9f ) ***
C** n+1 n n n-1 n-2 n-3 ***
C* * * * *
C** f -f are de f i n e d  by f = y ' (x ,y ) ***
C** n  n-3 n n n ***
C** ***
C** and are pas s e d  into A PR ED from the m a i n  p r o g r a m . ***
C** ***
C** This fourth ord er Predictor, w h e n  u s e d  w i t h  an app ropria te *** 
C** fourth order Corrector has a global truncation error on ***
C** the order of h A4 , so a step size smaller than IO e - 02 ***
C** will be aff ected largely by ma c h i n e  error and not by *■**
C** error in integration. ***
C** ***
C** ***
C* * ***
Q***********************************************************************
REAL F U N CTION A P R E D (Y n ,F n ,F n _ l ,F n _ 2 ,F n _ 3 )
IMPLICIT REAL*8 (A-H), (O-Z)
C O M M O N /IN T V A R S / T N O W rT F I N , T N R M L I Z D ,h ,h O V E R 2 4 
C O M M O N /I N T VARS2/ N _ s t _ e q s ,N _ b t w _ o u t ,N _ p f n t _ v a r (50)
A P R E D  = Yn + hOVER24* ( 5 5 *Fn - 59.*Fn_l + 37.*Fn_2 - 9.*Fn_3)
RET URN
END
Q * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *  
c** * * *
C** A C O R R  is a fourth order ADA M S - M O U L T O N  corrector to be u s e d  *** 
C** w i t h  a fourth order A D A MS-BAS HFORTH p re dictor A P R E D . ***
C** The corrector is of the form ***
C** ***
C** Y  =Y + h/24*(9f +19f -5f +f )
C** n+1 n n+1 n n-1 n-2C**
C** f -f are de f i n e d  by f = y ' (x ,y )
C** n n-2 n n n
C* *
* * *
* * *
* * *  
* * *
* * *
* * *
C* * 
(2* * 
(2* * 
Q**
C** 
c* * 
c** 
C**
c* *
c* *
an d  are pas s e d  into A CO RR from the m a i n  program. ***
This fourth order Corrector, w h e n  use d wi t h  an appropria te 
fourth order Predictor has a global truncation err or on 
the order of h A4 , so a step size smaller than 10e-02 
will be aff ected largely by m a c h i n e  error and not by 
error in integration.
* * * 
* * *  
* * * 
* * *  
* * *  
* * *
* * *
* * *
Q * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *
Figure 49. Continued.
152
REAL F U N CTION A C O R R (Y n ,F n p l u s l ,F n ,F n _ l ,F n _ 2 )
IMPLICIT REAL*8 (A-H)z (O-Z)
C O M M O N / IN T V A R S / TNO W zT F I N zT N R M L I Z D ,h ,hOVER2 4 
C O M M O N /I N T VARS2/ N_s t _ e g s ,N_btw_ou t > N _ p r n t _ v a r (50)
A C O R R  = Yn + h O V E R 2 4 * (9.*Fnplusl + 19.*Fn - 5.*Fn_l + F n _ 2 )
RET URN
END
Q*********************************************************************** 
c** ***
C** Sub routin e INPUT is u s e d  to define the us e r  parameters of the *** 
C** integration. The value for step size should be de f i n e d  a n d  ***
C** the number of state variables declared. The common var iable ***
C** hOVER24 should also be initialized. Also, ***
C** cer tain changeable inputs can be initialized here and us e d  ***
C** throughout the code such as in STATE. ***
C * *  * * *
c***************************************.************ *■* ******************
SUBROUTINE INPUT
IMPLICIT R E A L * 8 (A-H)z (O-Z)
REAL*8 L p z L s zL m zL z L i n v z LinvR
INTEGER N_ p r n t _ v a r , SaveTruncTNRML
D I M ENSION L (50,50),R (50,50)
C O M M O N / INTVARS/ T N O W zT F I N zT N R M L I Z D zh,h O V E R 2 4 
C O M M O N /INTVARS2/ N _ s t _ e q s ,N _ b t w _ o u t ,N _ p r n t _ v a r (50) 
COMMON/STVARS/ PERIOD,V s ,L i n v (50,50),LinvR(SOzSO)
C O M M O N / PAR A M S / a , R S l (50)zR S 2 p ( 5 0 ) zR S 2 s ( 5 0 ) ,R l ,S U P P L Y (50)
C O M M O N /P A R AMSI / L m zLs
C O M M O N /P A R A M S 2 / R S l c l o s e d zRS2popen
C
C
C
C
C
C
C
If this is the first time through, initialize the integration 
parameters. Otherwise update the L and R matrices.
I F (h .NE. 0.D O ) GO T O  7
Integration parameters
W R I T E (*,*) ' Input the number of state equations. <??>' 
R E A D (*,*) N_st_eqs
W R I T E (*,*) ' Input the finish time. <?.??>'
R E A D (*,*) TFIN
W R I T E (*,*) ' Input the step size. <?.??>'
R E A D (*,*) h 
hOVER2 4 = h/24.DO
Figure 49. Continued.
n
o
n
153
WRITEf*,*) ' Input the num ber of steps between calls to OUTPUT. 
& <??>'
R E A D (*,*) N_b tw_ou t
WRITEf*,*) ' Input the number of output variables. <??>'
R E A D (*,*) N _ p r n t _ v a r (I )
WRITEf*,*) ' Input the output variables. <??,??,??,...>' 
READf*,*) (N_ prnt_v ar(I), 1=2,N _ p r n t _ v a r (I )+1)
- U s e r  par ameter s
WRITEf*,*) ' Input the turns ratio. (Prim:Sec=a:I) <?.??>' 
READf*,*) a
WRITEf*,*) ' Input the primary inductance in H e n r y s . <?.??>' 
READf*,*) Lp
WRITEf*,*) ' Input the secondary inductance in H e n r y s . <?.??>' 
READf*,*) Ls
WRITEf*,*) ' Input the mag netizi ng inductance in H e n r y s . <?.??>' 
READf*,*) Lm
WRITEf*,*) ' Input the core loss resistance in Ohms. <?.??>' 
READf*,*) Re
WRITEf*,*) ' Input the source resistance in Ohms. <?.??>' 
READf*,*) R S l closed
WRITEf*,*) ' Input the open source GTO forward res istanc e 
&in O h m s . <?.??>'
READf*,*) RSlopen
WRITEf*,*) ' Input the primary sho rting (snubbing) 
^resistance in Ohms. <?.??>'
READf*,*) RS2 pclose d
WRITEf*,*) ' Input the primary open shorting (snubbing) 
SlG T O  resistance in O h m s . <?.??>'
REA D (*',*) RS2popen
WRITEf*,*) ' Input the secondary shorting (snubbing) 
Siresistance in O h m s . <?.??>'
READf*,*) RS2 sclose d
WRITEf*,*) ' Input the secondary open shorting (snubbing) 
SlGTO resistance in O h m s . <?.??>'
READf*,*) RS2sopen
WRITEf*,*) ' Input the load resistance in Ohms. <?.??>' 
READf*,*) Rl
WRITEf*,*) ' Input the period. <?.??>'
READf*,*) PERIOD
WRITEf*,*) ' Input the DC supply voltage. <?.??>.' 
R E A D (*,*) Vs
IFfVs .LT. 0.0) Vs = (- I .)*Vs
Figure 49. Continued.
n
o
n
 
n
n
n
n
n
n
n
 
n
o
n
154
- Initialize the m a t r i x  of inductances for the I-core.
D O  1=1,50 
DO J = I ,50
R (I ,J) = 0 .DO 
L (I, J) = 0 .DO 
L I I W ( I zJ) = 0 .DO 
L I N V R ( I zJ) = 0 .DO 
END DO 
END DO
Sav eTrunc TNRML = I
- V a r y  the Res istanc e values depending on w he ther or not the 
appropria te supplies are "switched in". This is sup posed 
to simulate diodes. Example: R S l = I .E+06 means that GTOl 
is open. This only needs to be done if this is the first 
time through, or a switch point has bee n reached.
I F (SaveTruncTNRML .EQ. IDI NT( T N R M L I Z D ) ) GO TO 77
DO 1=1,22,3
I F ( S U P P L Y d ) .EQ. 0 . D O )
I F ( S U P P L Y d ) .NE. 0 . D O )
I F ( S U P P L Y d ) .NE. 0. D O )
I F ( S U P P L Y d ) .EQ. 0. D O )
I F ( S U P P L Y d ) .NE. 0 . D O )
I F ( S U P P L Y d ) .EQ. 0. D O )
END DO
- D ev elop the m a t r i x  of
R S l ((1+2)/3) = RSlopen 
RSl ( (1+2)/3) = R S l c l o s e d
R S 2 p ( (1+2)/3) = RS2popen 
R S 2 p ((1+2)/3) = RS2pclosed
RS2s( (1+2)/3) = RS2sopen 
RS2s ( (1+2)/3) = RS2 sclose d
inductances for the 1-core.
L ( I zI) = Lp/a**2 
L (I,2) = Lm/a**2 
L (2,2) = L m /Re
L (3,2 ) = - L m * ( I . D 0 / R S 2 s (I)+ 1 .D0/R1)/a**2 
L (3,3) = L s * ( I . D 0 / R S 2 s (I )+1.D0/R1)/a**2 
L (3,5) = - L m / (Rl*a**2)
L ( 3 , 6) = Ls/Rl 
L (3,8) = - L m / (Rl*a**2)
L (3,9) = Ls/Rl
L ( 3 , 11) = - L m / (Rl*a**2)
L (3,12) = Ls/Rl 
L (3,14) = - L m / (Rl*a**2)
L (3,15) = Ls/Rl
L ( 3 , 17) = - L m / (Rl*a* *2)
L (3,18) = Ls/Rl 
L (3,20) = - L m / (Rl*a**2)
L (3,21) = Ls/Rl 
L (3,23) = - L m / (Rl*a**2)
L (3,24) = Ls/Rl
Figure 49. Continued.
155
L(4,4) = Lp/a**2 
1,(4,5) = Lm/a**2 
L (5,5) = Lm/Rc 
L (6,2) = - L m / (Rl*a**2)
L (6,3) = Ls/Rl
L (6,5) = - L m * (I .D 0 / R S 2 s (2)+1.D0/R1)/ a * *2 
L ( 6 , 6) = L s * ( l . D 0 / R S 2 s ( 2 ) + l . D 0 / R 1 )/a**2 
L (6,8) = - L m / (Rl*a**2)
L (6,9) = Ls/Rl 
L (6,11) = - L m / (Rl*a**2)
L (6,12) = Ls/Rl 
L (6,14) = - L m / (Rl*a**2)
L (6,15) = Ls/Rl 
L (6,17) = - L m / (Rl*a**2)
L (6,18) = Ls/Rl 
L (6,20) = - L m / (Rl*a**2)
L (6,21) = Ls/Rl 
L (6,23) = - L m / (Rl*a**2)
L (6,24) = Ls/Rl
L (7,7) = Lp/a**2 
L (7,8) = L m / a * *2 
L (8,8) = Lm/Rc 
L (9,2) = - L m / (Rl*a**2)
L (9,3) = Ls/Rl 
L (9,5) = - L m / (Rl*a**2)
L (9,6) = Ls/Rl
L (9,8) = - L m * (I .D 0 / R S 2 s (3)+1.D0/R1)/a**2 
L (9,9) = L s * ( I . D 0 / R S 2 S (3)+1.D0/R1)/a**2 
L (9,11) = - L m / (Rl*a**2)
L (9,12) = Ls/Rl
L ( 9 , 14) = - L m / (Rl*a**2)
L (9,15) = Ls/Rl 
L (9,17) = - L m / (Rl*a**2)
L (9,18) = Ls/Rl 
L (9,20) = - L m / (Rl*a**2)
L (9,21) = Ls/Rl 
L (9,23) = - L m / (Rl*a**2)
L (9,24) = Ls/Rl
L ( I O fIO) = L p / a**2 
L ( I O fIl) = Lm/a**2 
L ( I l fIl) = Lm/Rc 
L (12,2) = - L m / (Rl*a**2)
L (12,3) = Ls/Rl •
L (12,5) = - L m / (Rl*a**2)
L (12,6) = Ls/Rl 
L (12,8) = - L m / (Rl*a**2)
L (12,9) = Ls/Rl
L (12,11) = - L m * (I . D 0 / R S 2 s (4)+1.D0/R1)/a**2 
L (12,12) = L s * ( I . D 0 / R S 2 s (4)+1.D 0 / R 1 )/a**2 
L (12 f14) = - L m / (Rl*a**2)
L (12,15) = Ls/Rl 
L (12,17) = - L m / (Rl*a**2)
L (12,18) = Ls/Rl 
L (12,20) = - L m / (Rl*a**2)
L (12,21 ) = Ls/Rl 
L (12,23) = - L m / (Rl*a**2)
L (12,24) = Ls/Rl
Figure 49. Continued.
156
1,(13,13) = Lp/a**2 
L (13,14) = Lm/a**2 
L (14,14) = Lm/Rc 
1,(15,2) = -Lm/ (Rl*a**2)
L (15,3) = Ls/Rl 
L (15,5) = - L m / (Rl*a**2)
L (15,6) = Ls/Rl 
L (15,8) = - L m / (Rl*a**2)
L (15,9) = Ls/Rl 
L (15,11) = - L m / (Rl*a**2)
L (15,12) = Ls/Rl'
L (15,14) = - L m * ( I . D 0 / R S 2 S (5)+1.D0/R1)/a**2 
L (15,15) = L s * (I. D 0 / R S 2 s (5)+1.D 0 / R 1 )/a**2 
L (15,17) = - L m / (Rl*a**2)
L (15,18) = Ls/Rl 
L (15,20) = - L m / (Rl*a**2)
L (15,,21) = Ls/Rl 
L (15,23) = - L m / (Rl*a**2)
L (15,24) = Ls/Rl
L (16,16) = Lp/a**2 
L (16,17) = L m / a * *2 
L (17,17) = Lm/Rc 
L (18,2) = - L m / (Rl*a**2)
L (18,3 ) = Ls/Rl 
L (18,5) = - L m / (Rl*a**2)
L (18,6 ) = Ls/Rl 
L (18,8) = - L m / (Rl*a**2)
L (18,9) = Ls/Rl 
L (18,11) = - L m / (Rl*a**2)
L (18,12) = Ls/Rl 
L (18,14) = - L m / (Rl*a**2)
L (18,15) = Ls/Rl
L (18,17) = - L m * ( I . D 0 / R S 2 s (6)+1.D0/R1)/a**2 
L (18,18) = Ls* (l . D 0 / R S 2 s ( 6 ) + l . D 0 / R 1 )/a**2 
L (18,20) = - L m / (Rl*a**2)
L(18,21) = Ls/Rl 
L (18,23) = - L m / (Rl*a**2)
L (18,24) = Ls/Rl
L(19,19) 
L(19,20) 
L (20,20) 
L (21,2)
L (21,3)
L (21,5)
L (21,6)
L (21,8)
L(21,9)
L (21,11) 
L (21,12) 
L (21,14) 
L (21,15) 
L(21,17) 
L(21,18) 
L (21,2 0) 
L (21,21) 
L (21,23) 
L (21,2 4)
= Lp/a**2 
= Lm/a**2 
= Lm/Rc
= - L m / (Rl*a**2)
= Ls/Rl
= - L m / (Rl*a**2)
= Ls/Rl
= - L m / (Rl*a**2)
= Ls/Rl
= - L m / (Rl*a**2)
= Ls/Rl
= - L m / (Rl*a**2)
= Ls/Rl
= - L m / (Rl*a**2)
= Ls/Rl
= -Lm* (I .D0/RS2S(7)+l.D0/Rl)/a**2 
= L s * ( I . D 0 / R S 2 S (7)+1.D0/R1)/a**2 
= - L m / (Rl*a**2)
= Ls/Rl
Figure 49. Continued
157
L (22,22) = Lp/a**2 
L (22,23) = Lm/a**2 
L (23,23) = Lm/Rc- 
L (24,2) = - L m / (Rl*a**2)
L (24,3) = Ls/Rl 
L (24,5) = - L m / (Rl*a**2)
L (24,6) = Ls/Rl 
L (24,8) = - L m / (Rl*a**2)
L (24,9) = Ls/Rl 
L (24,11) = - L m / (Rl*a**2)
L (24,12) = Ls/Rl 
L (24,14) = - L m / (Rl*a**2)
L (24,15) = Ls/Rl 
L (24,17) = - L m / (Rl*a**2)
L (24,18) = Ls/Rl 
L (24,20) = - L m / (Rl*a**2)
L (24,21) = Ls/Rl
L (24,23) = - L m * ( I . D 0 /RS2S(8)+1.D0/R1)/a**2 
L (24,24) = L s * ( I . D 0 / R S 2 S (8)+1.D0/R1)/a**2
C - Invert L
CALL F U L PIVOT( L,24,LINV)
C - Develop the m a t r i x  of resistance coefficients for the 8-core.
R(l,l) = 
R ( 2 , I) 
R (2,2) = 
R (2,3 ) = 
R(3,3 ) =
- R S 2 p (I)* R S 1 (I)/ (a* *2*(RS I(I)+ R S 2p(l)))
= I . DO
-I. DO
-I. DO
-I. DO
R ( 4 , 4) = 
R<5,4) 
R ( 5 , 5) = 
R ( 5 , 6) = 
R ( 6, 6) =
-RS 2p(2) * R S l ( 2 ) / ( a * * 2 * ( R S l ( 2 ) + R S 2 p (2)))
= 1.D0
-I.DO
-I.DO
-I .DO
R(7,7) = 
R(8,7) 
R (8,8) = 
R(8,9) = 
R(9,9) =
- R S 2 p (3)* R S 1 (3)/(a* *2 * (RS I(3)+ R S 2 p (3)))
= 1.D0
-I.DO
-I.DO
-I.DO
R ( I O 1IO) = 
R ( I l 1IO) 
R ( I l 1Il) = 
R(11,12) = 
R (12,12) =
- R S 2 p (4)* RSl(4) /(a**2 *(RSl (4)+RS 2p(4)))
= I . DO
-I.DO
-I. DO
-I. DO
R ( I S 1IS) = 
R(14,13) 
R (14,14) = 
R (14,15) = 
R ( I S 1IS) =
- R S 2 p (5)* RSl(5) /(a**2 *(RSl (5)+RS 2p(5)))
= 1.D0
-I.DO
-I.DO
-I.DO
R ( I G 1IG) = 
R (17,1 6) 
R (17,17 ) = 
R (17,18) = 
R ( I S 1IS) =
-RS 2p( 6 ) * R S l ( 6 ) / ( a* *2*(RS l(6)+ R S 2 p (6)))
= I.DO
-I.DO
-I.DO
-I. DO
Figure 49. Continued
158
R (19,19) = 
R (20,19) 
R (20,20) = 
R(20,21) = 
R (21,21 ) =
- R S 2 p ( 7 ) * RSl(7) /(a**2 *(RSl (7)+RS 2p(7)))
= I .DO
-I.DO
-I. DO
-I.DO
R (22,22) = 
R(23,22) 
R (23,23) = 
R (23,24) = 
R (24,24) =
- R S 2 p (8)* R S 1 (8)/(a**2*(RSl(8)+RS2p(8)))
= I.DO
-I. DO
-I.DO
-I.DO
C - M u l tiply L i n v  and R
CALL E M P R O D (LINV,R ,24,24,24,L I N V R )
77 SaveTrunc TNRML = IDINT(TNRMLIZD)
RET URN
END
Q * * * * * * * . * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *
C
C M A T R X P A K  is a small set of routines that do the g r i n d i n g
C inv olved in numerical m a t r i x  c a l c u l a t i o n s . They are
C equivalent to addition/subtraction, m u l tiplic ation and 
C d i v ision for scalars. M u l tiplic ation has two a l g o r i t h m s :
C The gar d e n  variety procedure of forming rep eated inner 
C (dot) vec t o r  products, and an algorithm by Lynn Can n o n  that
C is suited to parallel pro cessin g in an array m ac hine that
C is nearest-n eighbo r connected, e . g . , ILLIAC IV. The inversion
C routine does full p i v oting of the array so that effects of
C numerical ill -conditioning are minimized. All routines show
C d im ension ing for 6 x 6  arrays but are useable for any size 
C p r o b l e m  without modification, since subroutine dim ension ing 
C indicates only how indexing is to be performed.
C
C The routines are not in way proprietary. Feel free to 
C transport them anywhere.
C
Q********************************************************************
Q************************************************
C Sub routin e EMPROD mul tiplie s mat rices 
C u s i n g  classical notions of inner products
SUBROUTINE E M P R O D (A,B,N, M , L,C)
R E A L *8 A , B ,C
DIM E N S I O N  A ( 5 0 , 5 0 ) , 8 ( 5 0 , 5 0 ) ,C(50,50)
DO 1=1,N
DO J = I zL 
C ( I zJ)=O.
DO K = I zM
C ( I zJ) = C ( I zJ) + A ( I zK ) * B (KzJ) 
END DO 
END DO 
END DO 
RETURN 
END
Figure 49. Continued.
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o**************************************************************** 
G FUL P I V O T  inverts mat rices u si ng a Gau ss-Jor dan pro c e d u r e
C w i t h  full-pivoting.
C
C A A  = M a t r i x  to be inverted
C B =  Inverse of A A
C N  = Size of mat rices A A  and B
C
C * ***************************** ********************************** 
SUBROUTINE F U L P IV O T (AA,N,B)
R E A L *8 AA , A , B
D I M E N S I O N  A A (50,50),A(50,50), 8 ( 50,50),JK(SO)
If A A  is only I x  I (a s c a l a r ) , invert and return.
IF (N .EQ. I) THE N
IF ( A A (1,1) .NE. 0.0) THEN 
B (I ,I ) = 1.0 / A A (1,1)
ELSE
W R I T E (7,100)
O FORMAT (/' » » >  Determinant is zero.',
& ' No inversion. « « ' / )
END IF 
RET URN  
END IF
Cop y A A  into a wo r k i n g  array. At the same time, construct an 
identity matrix, B, into which row operations on A  will be 
copied, thus forming the inverse. Als o clear the J K  array 
(which is u s e d  to m a r k  those rows that have been processed) 
to indicate that no rows have been processed.
DO 1=1,N
DO J = I ,N
A ( I zJ) = A A ( I zJ)
B ( I zJ) = 0.0 
END DO 
END DO
DO I = I zN
B ( I zI) = 1.0 
JK(I) = O  
END DO
Execute the reduction loop N  times 
DO K = I fN
C Fin d the pivot element. It is the largest element in the array 
C that is in the rows and columns that have not been processed.
C The che ck for noh -proce ssed rows is for JK = 0.
C
COMP = 0.0
Figure 49. Continued.
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DO J = I jN
IF (JK(J) .EQ. 0) THEN 
DO I = I zN
IF (JK(I) .EQ. 0) THEN
IF (AB S( A ( I zJ) ) .GT. COMP) THE N
COMP = A B S ( A ( I zJ) )
II = I 
JJ = J 
END IF 
END IF 
END DO 
END IF 
END DO 
JK(JJ) = I
C O M P = O .
DO 4 J = I zN  
I F (J K ( J ) ) 4 , 5 z 4
5 DO 14 I = I zN  
I F ( J K d )  )14 , 1 5 r14
15 IF (COMP-ABS (A (I, J) ) ) 6,14,1-4
6 C O M P = A B S ( A d zJ) )
II=I
JJ= J
14 CON TINUE 
4 CON TINUE 
JK(JJ)=I
Interchange rows if necessary. Ma k e  the same interchanges on A  and B .
IF (II .NE. J J ) THEN 
DO J = I zN
TEMP = A ( I I zJ)
A ( I I zJ) = A ( J J zJ)
A ( J J zJ) = TEMP 
TEMP = B ( I I zJ)
B ( I I zJ) = B ( J J zJ)
B ( J J zJ) = TEMP 
END DO 
END IF
Nor m a l i z e  the pivot row.
IF ( A ( J J zJJ) .EQ. 0.0) THEN 
W RI TE (7,, 100)
RETURN 
END IF
TEMP = 1.0 / A ( J J zJJ)
DO J = I zN
A ( J J zJ) = TEMP * A ( J J zJ)
B ( J J zJ) = TEMP * B ( J J zJ)
END DO
A ( J J zJJ) = 1.0
Figure 49. Continued
161
C Per form the ro w  operations. Rec ord them in bot h A  and B . 
C
DO 1=1,N
IF (I .NE. JJ) THE N  
TEMP = A ( I fJJ)
DO J = I fN
A ( I fJ) = A ( I fJ) - T E M P * A (JJfJ)
B ( I fJ) = B ( I fJ) - T E M P * B (JJfJ)
END DO 
END IF 
END DO
END DO 
RET URN  
END
C******************** ***** ****** ****** ****** ***** ****** ****** ****** *****
C** ***
C** Sub routin e INITIAL is u s e d  to find the values of the sys tem at *** 
C** the first three integation s t e p s . Usually, a Run ge-Kut ta ***
C** routine w o u l d  be u s e d  to find these values, but it ma y  be ***
C** pos sible to assume values for these states if the system can *** 
C** be a ss umed to be at rest for some time before the integration*** 
C** is to start. ***
C** ***
C** Yn_3 is known (initial condition) ***
C** F n _ 3 , F n _ 2 , F n _ l , and Yn nee d to be found by initial. ***
C* * ***
Q * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *
SUBROUTINE I N I T I A L (Yn,F n _ l ,F n _ 2 ,F n _ 3 )
IMPLICIT R E A L * 8 (A-H), (O-Z)
DIM E N S I O N  Y n ( I O O ) ,F n _ l (100),F n _ 2 (100),F n _ 3 (100)
DO 1=1,100
Yn(I) = 0.0
F n _ l (I ) = 0.0
F n _ 2 (I) = 0.0
F n _ 3 (I) = 0.0
END DO
RETURN
END
Q************************* ****** ****** ****** ***** ****** ****** ****** *****
Q * *  * * *
C** Sub routin e STATE is a use r-writ ten subroutine . ***
C** that describes the state equations for continuous ***
C** m o d e l i n g  of a system. In this case, the system is ***
C** the 8-core PASC transformer w i t h  series connected ***
C** s e c o n d a r i e s . The state equations are of the form ***
C** . ***
C** Lx = R x  + u ***
C** w he re L is the m a t r i x  of inductances, R is the m a t r i x  ***
C** of resistances and u is a vec tor of driving functions. ***
c* * ■ * * *.
C** ' ***
Q * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *
Figure 49. Continued.
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SUB ROUTIN E S T A T E (DD,S S )
C - Dec lare variables that aren't implicit and dimension arrays
IMPLICIT REAL*8 (A-H), (O-Z)
R E A L *8 L I N V (LIN VR,PER IOD,Vs
INTEGER L O C K (20)
D I M E N S I O N  DD(IOO),SS(IOO)
- Common buf f e r  STVARS is a buffer from input that contains 
some state equ ation parameters 
INTVARS is the integration variables buffer
C O M M O N / IN T V A R S / T N O W ,TEIN,T N R M L I Z D ,h ,hOV E R 2 4 ■
C O M M O N /INT VARS2/ N_st_eqs,,N_btw_out , N_p rnt_va r (50) 
C O M M O N / S T V A R S / PERIOD,V s ,L l n v (50,50),L i n v R (50,50)
C O M M O N / P A R A M S / a,R S l ( S O ) ,R S 2 p (50),R S 2 s (50),R l ,S U P P L Y (50) 
C O M M p N /PAR AMS2/ RSl closed ,RS2popen
- Determine w h i c h  DC supplies are e ne rgized by finding out 
w he re y o u  are in time relative to one period.
DO 1=1,50
SUPPLY(I) = 0.D0 
END DO
T N R M L IZD = D M O D (TNOW,P ER IOD)*32.DO/PERIOD
I F ((TNRMLIZD .GE. FLOAT(I)) .AND.
Sc (TNRMLIZD .L T . FLOAT(S) ) ) THE N
S U P P L Y (I) = Vs/ a*RS2popen/(RSlclosed+RS2popen)
END IF
IF((TNRMLIZD .GE. F L O A T (2)) .AND.
Sc (TNRMLIZD .L T . FLOAT(IO) ) ) THEN
S U P P L Y (4) = Vs/ a*RS2popen/(RSlclosed+RS2popen)
END IF
IF((TNRMLIZD .GE. F L O A T (3)) .AND.
Sc (TNRMLIZD . L T . FLOAT(Il) ) ) THEN
S U P P L Y (7) = Vs/ a*RS2popen/(RSlclosed+RS2popen)
END IF
IF((TNRMLIZD .GE. F L O A T (4)) .AND.
Sc (TNRMLIZD . L T . F L O A T (12) ) ) THEN
S U P P L Y (10) = Vs/a*RS2popen/(RSlclosed+RS2popen)
END .IF
IF C(TNRMLIZD .GE. FLOAT(S)) .AND.
Sc (TNRMLIZD . L T . F L O A T (13) ) ) THEN
S U P P L Y (13) = Vs/a*RS2popen/(RSlclosed+RS2popen)
END IF
I F ((TNRMLIZD .GE. FLOAT(S)) .AND.
Sc (TNRMLIZD . L T . F L O A T (14))) THEN
S U P P L Y (16) = Vs/a*RS2popen/(RSlclosed+RS2popen)
END IF
I F ((TNRMLIZD .GE. F L O A T (7)) .AND.
Sc (TNRMLIZD . L T . FLOAT(IS) ) ) THEN
S U P P L Y (19) = Vs/a*RS2popen/(RSlclosed+RS2popen)
END IF
I F ((TNRMLIZD .GE. FLOAT(S)) .AND.
Sc (TNRMLIZD .L T . FLOAT(IS) ) ) THE N
S U P P L Y (22) = Vs/a*RS2popen/(RSlclosed+RS2popen)
END IF
Figure 49. Continued.
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I F ((TNRMLIZD .GE. FLOAT(IV)) .AND.
& (TNRMLIZD . L T . F L O A T (25))) THE N
S U P P L Y (I) = -Vs/a*RS2popen/(RSlclosed+RS2popen) 
END IF
IF((TNRMLIZD .GE. F L O A T ( I S ) ) .AND.
& (TNRMLIZD .L T . F L O A T (26))) THEN
S U P P L Y (4) = -Vs/a*RS2popen/(RSlclosed+RS2popen) 
END IF
IF((TNRMLIZD .GE. F L O A T ( I P ) ) .AND.
& (TNRMLIZD . L T . F L O A T (27))) THE N
SUPPLY (7) = -Vs/a*RS2popen/ (RSlclosed+R'S2popen) 
END IF
IF((TNRMLIZD .GE. F L O A T (20)) .AND.
Sc (TNRMLIZD .L T . F L O A T (28) ),) THE N
S U P P L Y (10) = -Vs/a*RS2popen/(RSlclosed+RS2popen) 
END IF
IF((TNRMLIZD .GE. F L O A T ( 2 1 ) ) .AND.
Sc (TNRMLIZD .LT. F L O A T (29) ) ) THE N
S U P P L Y (13) = -Vs/a*RS2popen/(RSlclosed+RS2popen) 
END IF
IF((TNRMLIZD .GE. F L O A T (22)) .AND.
Sc (TNRMLIZD .LT. F L O A T (30) ) ) THEN
S U P P L Y (16) = -Vs/a*RS2popen/(RSlclosed+RS2popen) 
END IF
IF((TNRMLIZD .GE. F L O A T (23)) .AND.
Sc - (TNRMLIZD .LT. FLOAT(31) ) ) THEN
S U P P L Y (19) = -Vs/a*RS2popen/(RSlclosed+RS2popen) 
END IF
IF((TNRMLIZD .GE. F L O A T (24)) .AND.
Sc (TNRMLIZD .LT. FLOAT (32) ) ) THEN
S U P P L Y (22) = -Vs/a*RS2popen/(RSlclosed+RS2popen) 
END IF
- Call input to update the LIN V and LIN VR matrices due to time
v ar ying resistances.
CALL INPUT
- W r i t e  the state equations.
D D (i) is the vec t o r  of time derivatives of current x  
S S (i) is the vec t o r  of currents x  
SS(I) is the first pos itive primary (PO)
LIN V is the inverse of the inductance mat rix
LIN VR is the above mul t i p l i e d  by the resistance m a t r i x
SUPPLY is the vec tor of DC supply voltages
DO 11 I = 1,24 
DD(I) = 0.DO 
DO 10 J = 1,24
DD(I) = DD(I) + L I N V R (I ,J ) *SS(J) + L I N V (I , J ) * S U P P L Y (J)
10 CON TINUE
11 CON TINUE
C - Ret urn to M A I N  
RETURN 
END
Figure 49. Continued.
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c******************************************************* **********.,.***** 
C** * * *
C** Sub routin e OUT P U T  is u s e d  to format the integration o u t p u t . ***
C** * * *
Q***********************************************************************
SUB ROUTIN E O U T P U T (YnzFn)
IMPLICIT REAL*8 (A-H), (O-Z)
R E A L *8 L m zL s zIsec
i
INTEGER N_p rnt_va r,N_pl ot 
D I M E N S I O N  Yn(IOO),Fn(IOO)
C O M M O N /INTVARS Z T N O W , T F I N z T N R M L IZ D ,h ,hOVER2 4 
COM MON/INTVARS2Z N _ s t _ e q s , N _ b t w _ o u t , N _ p r n t _ v a r (50) 
C O M M O N / PARAMSZ a , R S l (50)zR S 2 p ( 5 0 ) zR S 2 s (50),R l ,S U P P L Y (50) 
C O M M O N /PAR A M S I / L m zLs
- This is T EK PLOT format
I F (TNOW .EQ. 0.0) THE N
W R I T E (I,*) 'US Dept, of Energy PASC Project' 
W R I T E (I,*) ' '
- Adj ust for add ition of Isec to output plots
N_plot = N _ p r n t _ v a r (I) + I 
W R I T E (I,102) N_plot 
END IF
- Secondary current is same as 1st primary unless R S 2 s (I) is 
shorting. The n use current in last primary.
Isec = 0 .DO
IF ( R S 2 s (I) .GE. 1.D3) THEN 
Isec = Y n (3)
ELSE
Isec = Yn(24)
END IF
W R I T E ( I zIOI) T N O W z Is e c ,(Y n (N_prnt_v ar(I )), 1 = 2 zN _ p r n t _ v a r (I)+1)
101 F O R M A T ( I X , F 9 . 5 , 8 (2X,',',F9 .5))
102 F O R M A T d X z 12)
RET URN
END
Figure 49. Continued.
165
Figure 50 shows a file that can be used to execute the 
integration routine along with some typical measured values 
for the PASC system.
$!
$!
$!
$!
$!
$!
$!
$
- Com file to run PASC.EXE on a VAX/VMS system.
SET DEF myhome 
RUN/nodebug PASC
24 ; #
5.D-Ol
l.D-07
500 ; #
6 ; #
1,2,3,16,17,18 
5.D-Ol 
2.3595D-04 
9.438D-04 
I.5075D-01 
20.DO 
10.DO
1. D4
2. d-02 
1.D4 
l.d6 
1.D6 
100.DO
I.6666667D-02 
60 .DO 
t4.dat
$! del t4.dat;* 
$!
$ EXIT
of state variables 
,•Finish time 
;Delta T
steps between output calls 
of output variables
' ;output varibles 
;turns ratio 
;Lp 
;Ls 
;Lm 
; Re
;RSlclosed 
;RSlopen 
;RS2pclosed 
;RS2popen 
;RS2sclosed 
;RS2sopen 
; Rl
;Period 
,-Vsource
Figure 50. File to Execute the Integration Routine of Fig­
ure 49.
762 100972363